hcju/isabelleps huggingface dataset:
CFGExit_def: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b" shows "CFGExit sourcenode targetnode kind valid_edge Entry Exit \<equiv> CFG sourcenode targetnode valid_edge Entry \<and> CFGExit_axioms sourcenode targetnode kind valid_edge Entry Exit"
aa: fixes j :: "nat"   and k :: "nat" assumes "j < k" shows "a j \<lless> a k"
is_unit_normalize: fixes a :: "'a" assumes "is_unit a" shows "normalize a = (1::'a)"
pair_bidirected_digraphI_bidirected_digraph: fixes G :: "'b pair_pre_digraph" assumes "bidirected_digraph (with_proj G) (swap_in (parcs G))" shows "pair_bidirected_digraph G"
simps_5: fixes f1 :: "'e"   and f2 :: "'f \<Rightarrow> 'f list \<Rightarrow> 'e"   and x21 :: "'f"   and x22 :: "'f list" shows "(case x21 # x22 of [] \<Rightarrow> f1 | x # xa \<Rightarrow> f2 x xa) = f2 x21 x22"
size_of_mat_R: fixes signs :: "rat list list"   and subsets :: "(nat list \<times> nat list) list" shows "M_mat_R signs subsets \<in> carrier_mat (length subsets) (length signs)"
isomorphicE: fixes a :: "'a \<times> 'a"   and a' :: "'a \<times> 'a" assumes "VV.isomorphic a a'" obtains f :: "'a \<times> 'a" where "VV.in_hom f a a' \<and> VV.iso f"
linear_order_on_def: fixes A :: "'a set"   and r :: "('a \<times> 'a) set" shows "linear_order_on A r \<equiv> partial_order_on A r \<and> Relation.total_on A r"
range_composition: fixes f :: "'d::type \<Rightarrow> 'b::type"   and g :: "'c::type \<Rightarrow> 'd::type" shows "range (\<lambda>x::'c::type. (f::'d::type \<Rightarrow> 'b::type) ((g::'c::type \<Rightarrow> 'd::type) x)) = f ` range g"
keys_row_to_poly: fixes ts :: "'d list"   and r :: "'e vec" shows "keys (row_to_poly ts r) \<subseteq> set ts"
nvalid_def2: fixes n :: "nat"   and P :: "assn"   and c :: "stmt"   and Q :: "assn" shows "\<Turnstile>n: (P, c, Q) \<equiv> \<forall>s t. s -c-n\<rightarrow> t \<longrightarrow> P s \<longrightarrow> Q t"
b_least2_scheme: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat"   and g :: "nat \<Rightarrow> nat"   and h :: "nat \<Rightarrow> nat" assumes "f \<in> PrimRec2"   and "g \<in> PrimRec1"   and "\<forall>x. h x < g x"   and "\<forall>x. f x (h x) \<noteq> 0"   and "\<forall>z x. z < h x \<longrightarrow> f x z = 0" shows "h \<in> PrimRec1"
mIH: shows "generalized_sfw (generalized_fw_join as ((bm, bd) # bs)) p = generalized_sfw (generalized_fw_join as bs) p"
C_Hole: shows "(\<lambda>t. t) \<in> ctxt"
eq_for_lvar_code_def: shows "eq_for_lvar_code \<equiv> EqForLVar.eq_for_lvar eq_idx_for_lvar"
D: shows "1 - alpha \<le> PS.prob {x \<in> space M. \<forall>k<K. \<bar>g x + birkhoff_sum f k x - g ((T ^^ k) x)\<bar> \<le> D}"
eSwapAbs_simp3: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'd, 'c) model"   and zs :: "'varSort"   and z1 :: "'var"   and z2 :: "'var" shows "eSwapAbs MOD zs z1 z2 ERR = ERR"
strict_mono_def: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'c" assumes "class.order less_eq less" shows "order.strict_mono less f = (\<forall>x y. less x y \<longrightarrow> f x < f y)"
full_functor_def: fixes A :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and B :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and F :: "'a \<Rightarrow> 'b" shows "full_functor A B F \<equiv> functor A B F \<and> full_functor_axioms A B F"
pderiv_set_flatten_PLUS: fixes X :: "'a rexp set"   and a :: "'a" assumes "finite X" shows "pderiv a (flatten PLUS X) = pderiv_set a X"
sets_PiM_I: fixes J :: "'b set"   and I :: "'b set"   and E :: "'b \<Rightarrow> 'c set"   and M :: "'b \<Rightarrow> 'c measure" assumes "finite J"   and "J \<subseteq> I"   and "\<forall>i\<in>J. E i \<in> sets (M i)" shows "prod_emb I M J (Pi\<^sub>E J E) \<in> sets (Pi\<^sub>M I M)"
function_ring_car_eqI: fixes R :: "('c, 'd) ring_scheme"   and f :: "'e \<Rightarrow> 'c"   and S :: "'e set"   and g :: "'e \<Rightarrow> 'c" assumes "ring_functions R"   and "f \<in> carrier (function_ring S R)"   and "g \<in> carrier (function_ring S R)"   and "\<And>a. a \<in> S \<Longrightarrow> f a = g a" shows "f = g"
long_division_closed_2: fixes K :: "('c multiset \<Rightarrow> 'a) set"   and p :: "('c multiset \<Rightarrow> 'a) list"   and q :: "('c multiset \<Rightarrow> 'a) list" assumes "subfield K L"   and "p \<in> carrier (K [X]\<^bsub>L\<^esub>)"   and "q \<in> carrier (K [X]\<^bsub>L\<^esub>)" shows "L.pmod p q \<in> carrier (K [X]\<^bsub>L\<^esub>)"
nn_integral_pmf: fixes A :: "'a set"   and p :: "'a pmf" shows "\<integral>\<^sup>+ x. ennreal (pmf p x) \<partial>count_space A = emeasure (measure_pmf p) A"
simps: fixes s1 :: "('e, 'f, 'g, 'h) SLin_state_scheme"   and s2 :: "('i, 'f, 'j, 'k) SLin_state_scheme" shows "P6 (s1, s2) = (\<forall>p. (status s1 p \<noteq> Aborted) = (status s2 p = Sleep))"
that: fixes T :: "'a set" assumes "T \<subseteq> \<V>"   and "int (card T) = \<t>"   and "finite T" shows "thesis"
treach_ref_l: fixes l :: "Location" assumes "ref l \<noteq> nullV" shows "treach \<tau> ref l (ltype l)"
dist_eq_0_iff: fixes x :: "'b"   and y :: "'b" shows "(dist x y = 0) = (x = y)"
hyps_2: fixes r :: "'a llist" assumes "l' = r @@ s" shows "r \<in> A\<^sup>\<star>"
N1_def: shows "N1 \<equiv> alpha1 \<or> beta1 \<or> gamma1"
not_outside_2: fixes p :: "'a pmf"   and q :: "'a pmf" assumes "p \<succeq>[relation] q" shows "q \<in> \<P> outcomes"
set_list_set: fixes xl :: "'a list" shows "set (list (set xl)) = set xl"
t_def: shows "t = (\<lambda>i. tl (m i))"
tendsto_MInfty: fixes f :: "'b \<Rightarrow> ereal"   and F :: "'b filter" shows "(f \<longlongrightarrow> - \<infinity>) F = (\<forall>r. \<forall>\<^sub>F x in F. f x < ereal r)"
fw_out_of_bounds2: fixes n :: "nat"   and j' :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and i' :: "nat" assumes "n < j'" shows "fw M n k i' j' = M i' j'"
ideal_points_on_unit_circle: fixes H :: "circline" shows "\<forall>z\<in>ideal_points H. z \<in> circline_set unit_circle"
inf2: fixes B :: "'c"   and k :: "'c"   and b :: "'c"   and a :: "'c" assumes "B \<equiv> k \<sqinter> b" shows "a \<sqinter> B \<equiv> k \<sqinter> (a \<sqinter> b)"
build_split: fixes w :: "nat \<Rightarrow> 'b" shows "w = w 0 ## suffix 1 w"
ia: shows "column ia (fst (QR_decomposition A)) \<in> columns (fst (QR_decomposition A))"
bound2b: shows "real_of_int (sint NEG_INF) < real_of_int (sint w2)"
simps_2: fixes A :: "'at" shows "positive (Neg A) = False"
less_cSUP_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 A :: "'c set"   and f :: "'c \<Rightarrow> 'b"   and a :: "'b" assumes "class.conditionally_complete_linorder Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq (f ` A)" shows "less a (Sup (f ` A)) = (\<exists>x\<in>A. less a (f x))"
sorted_map_suc: fixes l :: "nat list" assumes "sorted l" shows "sorted (map Suc l)"
xs: shows "S = {x} \<union> S' \<and> x \<notin> S'"
possExistenceImpliesNecEx: shows "\<lfloor>\<^bold>\<diamond>(\<lambda>v. \<exists>x. \<forall>xa. \<P> xa v = (\<^bold>\<box>xa x) v) \<^bold>\<rightarrow> \<^bold>\<box>(\<lambda>v. \<exists>x. existsAt x v \<and> (\<forall>xa. \<P> xa v = (\<^bold>\<box>xa x) v))\<rfloor>"
transpose_scalar: fixes k :: "real"   and A :: "(('a, 'c) vec, 'b) vec" shows "Finite_Cartesian_Product.transpose (k *\<^sub>R A) = k *\<^sub>R Finite_Cartesian_Product.transpose A"
mat_append_col_col: fixes A :: "'a mat"   and b :: "'a vec" assumes "dim_row A = dim_vec b" shows "col (mat_append_col A b) (dim_col A) = b"
correct: shows "\<exists>cnv. \<forall>\<D>. (\<forall>\<eta>. \<I>_ideal \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta> \<turnstile>g \<D> \<eta> \<surd>) \<longrightarrow> (\<forall>\<eta>. interaction_any_bounded_by (\<D> \<eta>) (bound \<eta>)) \<longrightarrow> (\<forall>\<eta>. lossless \<longrightarrow> plossless_gpv (\<I>_ideal \<eta> \<oplus>\<^sub>\<I> \<I>_common \<eta>) (\<D> \<eta>)) \<longrightarrow> (\<forall>\<eta>. wiring (\<I>_ideal \<eta>) (\<I>_real \<eta>) (cnv \<eta>) (w \<eta>)) \<and> Negligible.negligible (\<lambda>\<eta>. advantage (\<D> \<eta>) (ideal_resource \<eta>) (cnv \<eta> |\<^sub>= 1\<^sub>C \<rhd> real_resource \<eta>))"
lfp_trans_lemma2: fixes T :: "(('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real) \<Rightarrow> ('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real"   and v :: "('a \<Rightarrow> real) \<Rightarrow> 'a \<Rightarrow> real" assumes "\<And>t u. \<lbrakk>le_trans t u; \<And>P. sound P \<Longrightarrow> sound (t P); \<And>P. sound P \<Longrightarrow> sound (u P)\<rbrakk> \<Longrightarrow> le_trans (T t) (T u)"   and "\<And>t P. \<lbrakk>\<And>Q. sound Q \<Longrightarrow> sound (t Q); sound P\<rbrakk> \<Longrightarrow> sound (T t P)"   and "le_trans (T v) v"   and "\<And>P. sound P \<Longrightarrow> sound (v P)" shows "le_trans (T (lfp_trans T)) (lfp_trans T)"
inf_sup_ord_2: fixes x :: "'a multiset"   and y :: "'a multiset" shows "x \<inter># y \<subseteq># y"
elims: fixes x :: "bool list \<Rightarrow> bool"   and xa :: "bool list"   and xb :: "tree"   and y :: "tree" assumes "cutoff x xa xb = y" obtains red :: "bool list \<Rightarrow> bool" and ds :: "bool list" and T\<^sub>1 :: "tree" and T\<^sub>2 :: "tree" where "x = red__" and "xa = ds__" and "xb = Branching T\<^sub>1 T\<^sub>2" and "y = (if red__ ds__ then Leaf else Branching (cutoff red__ (ds__ @ [Left]) T\<^sub>1) (cutoff red__ (ds__ @ [Right]) T\<^sub>2))"   | red :: "bool list \<Rightarrow> bool" and ds :: "bool list" where "x = red" and "xa = ds" and "xb = Leaf" and "y = Leaf"
obj_simps_4: fixes a :: "'a" assumes "B.obj a" shows "B.dom a = a"
fps_lr_inverse_eq0_imp_starting0_2: fixes f :: "'a fps"   and x :: "'a" assumes "fps_right_inverse f x = 0" shows "x = (0::'a)"
bf_ite_def: fixes i :: "'a \<Rightarrow> bool"   and t :: "'a \<Rightarrow> 'b"   and e :: "'a \<Rightarrow> 'b" shows "bf_ite i t e \<equiv> \<lambda>l. if i l then t l else e l"
comp_arr_inv': fixes f :: "'b" assumes "\<phi>.B.iso f" shows "f \<cdot>\<^sub>B \<phi>.B.inv f = \<phi>.B.cod f"
of_nat_mod: fixes divide :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and minus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and one :: "'e"   and modulo :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and euclidean_size :: "'e \<Rightarrow> nat"   and division_segment :: "'e \<Rightarrow> 'e"   and m :: "nat"   and n :: "nat" assumes "class.unique_euclidean_semiring_with_nat divide plus minus zero times one modulo euclidean_size division_segment" shows "semiring_1.of_nat one plus zero (m mod n) = modulo (semiring_1.of_nat one plus zero m) (semiring_1.of_nat one plus zero n)"
filter_append: fixes P :: "'e \<Rightarrow> bool"   and xs :: "'e list"   and ys :: "'e list" shows "filter P (xs @ ys) = filter P xs @ filter P ys"
IH_1: fixes i :: "nat" assumes "MFOTL.future_reach \<phi>1 \<noteq> \<infinity>" shows "\<exists>j. i \<le> Monitor.progress \<sigma> \<phi>1 j"
exec_gpv_parametric': fixes S :: "'g \<Rightarrow> 'j \<Rightarrow> bool"   and CALL :: "'h \<Rightarrow> 'k \<Rightarrow> bool"   and R :: "'i \<Rightarrow> 'l \<Rightarrow> bool"   and A :: "'m \<Rightarrow> 'n \<Rightarrow> bool" shows "((S ===> CALL ===> rel_spmf (rel_prod R S)) ===> rel_gpv'' A CALL R ===> S ===> rel_spmf (rel_prod A S)) exec_gpv exec_gpv"
det_nfa_reach': fixes bd :: "nat option bdd"   and ls :: "bool list list"   and q1 :: "bool list"   and i :: "nat"   and q2 :: "bool list"   and j :: "nat"   and bss :: "bool list list" assumes "subset_dfs A (nfa_startnode A) = (bd, ls)"   and "\<exists>bs. nfa_reach A (nfa_startnode A) bs q1 \<and> list_all (is_alph n) bs"   and "q1 = ls ! i"   and "q2 = ls ! j"   and "i < length ls"   and "j < length ls"   and "list_all (is_alph n) bss" shows "nfa_reach A q1 bss q2 = (dfa_reach (det_nfa A) i bss j \<and> nfa_is_node A q2)"
S_alt_def: shows "S = {(l, u). l \<in> L \<and> u \<in> V \<and> u \<turnstile> PTA.inv_of A l}"
le_iff_meet: fixes x :: "'a"   and y :: "'a" assumes "x \<in> carrier L"   and "y \<in> carrier L" shows "(x \<sqsubseteq> y) = (x \<squnion> y = y)"
comp_by_index_inj: fixes x1 :: "nat"   and y1 :: "nat"   and x2 :: "nat"   and y2 :: "nat" assumes "comp_by_index x1 y1 = comp_by_index x2 y2" shows "x1 = x2 \<and> y1 = y2"
equal_ccsubspace_def: fixes A :: "'a ccsubspace"   and B :: "'a ccsubspace" shows "equal_class.equal A B = (A = B)"
NMPDA: fixes p :: "((int \<times> int) set set, int) Combinators list" assumes "DenyAll \<in> set p" shows "not_MT C p"
rel_mono: fixes R1 :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R1a :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and R2 :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and R2a :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "R1 \<le> R1a"   and "R2 \<le> R2a" shows "rel_optionT' R1 R2 \<le> rel_optionT' R1a R2a"
intro: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e" assumes "\<And>a b c. f (f a b) c = f a (f b c)" shows "semigroup f"
hyps_1: shows "q \<rightarrow> l qs \<in> \<delta>"
convex_rel_frontier_aff_dim: fixes S1 :: "'b set"   and S2 :: "'b set" assumes "convex S1"   and "convex S2"   and "S2 \<noteq> {}"   and "S1 \<subseteq> rel_frontier S2" shows "aff_dim S1 < aff_dim S2"
subgroup_is_group: assumes "Group.group G" shows "Group.group (G\<lparr>carrier := H\<rparr>)"
preserves_seq: fixes g :: "'c local.term set"   and f :: "'c local.term set" assumes "\<F>C.seq g f" shows "D.seq (E.map g) (E.map f)"
am6_lem: fixes y :: "('b \<times> 'b) set"   and x :: "('b \<times> 'b) set" shows "rel_ar y O x O rel_ar (y O x) = x O rel_ar (y O x)"
gfp_const: fixes t :: "'d" shows "(\<mu> x \<bullet> t) = t"
rtrancl_nderiv_nderivs: fixes A :: "'a set" shows "{((r, s), nderiv a r, nderiv a s) |r s a. a \<in> A}\<^sup>* = {((r, s), foldl (\<lambda>r a. nderiv a r) r w, foldl (\<lambda>r a. nderiv a r) s w) |r s w. w \<in> lists A}"
Col_def: fixes a :: "hyp2"   and b :: "hyp2"   and c :: "hyp2" shows "hyp2.Col a b c \<equiv> B\<^sub>K a b c \<or> B\<^sub>K b c a \<or> B\<^sub>K c a b"
assms: shows "moebius_pt M \<infinity>\<^sub>h = \<infinity>\<^sub>h"
groupF_equality: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list" shows "\<forall>A\<in>set (groupF f xs). \<forall>a1\<in>set A. \<forall>a2\<in>set A. f a1 = f a2"
path_to_then_lookup_Some: fixes s :: "nat \<Rightarrow> nat option"   and \<pi>s :: "char list list" assumes "\<exists>s'\<in>G. path_to s \<pi>s s'" shows "\<forall>\<pi>\<in>set \<pi>s. lookup_operator \<pi> \<noteq> None"
left_neutral: fixes a :: "'c" shows "(1::'c) \<cdot> a = a"
lower_power_float_interval: fixes p :: "nat"   and n :: "nat"   and x :: "float interval" shows "lower (power_float_interval p n x) = fst (float_power_bnds p n (lower x) (upper x))"
diamond_top_edges_def: shows "diamond_top_edges = (- 1, \<lambda>x. (diamond_x x, diamond_y (diamond_x x)))"
subformulas_in_subformulas_4: fixes G :: "'a formula"   and F :: "'a formula" assumes "\<^bold>\<not> G \<in> set (subformulae F)" shows "G \<in> set (subformulae F)"
blit_rule: fixes si :: "nat"   and len :: "nat"   and lsrc :: "'b list"   and di :: "nat"   and ldst :: "'b list"   and src :: "'b array"   and dst :: "'b array" assumes "si + len \<le> length lsrc"   and "di + len \<le> length ldst" shows "<src \<mapsto>\<^sub>a lsrc * dst \<mapsto>\<^sub>a ldst> blit src si dst di len <\<lambda>_. src \<mapsto>\<^sub>a lsrc * dst \<mapsto>\<^sub>a (take di ldst @ take len (drop si lsrc) @ drop (di + len) ldst)>"
GSMP_Union: fixes A :: "('b strand_label \<times> ('fun, 'var) strand_step) list" shows "GSMP (trms\<^sub>l\<^sub>s\<^sub>t A) = (\<Union>l. GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj l A)))"
aux: fixes n :: "nat"   and m :: "nat" assumes "odd n"   and "odd m" shows "(n div 2 = m div 2) = (m = n)"
derivg': fixes x :: "real" assumes "x \<in> {a..b}" shows "(g has_vector_derivative g' x) (at x)"
edge_succ_permutes: shows "edge_succ GM permutes arcs G"
RightTo_def: fixes a :: "e"   and \<phi> :: "e \<Rightarrow> c \<Rightarrow> w \<Rightarrow> bool" shows "RightTo a \<phi> \<equiv> \<lambda>c w. ob (pv w) (\<lambda>w. \<forall>x. (\<^bold>\<not>InterferesWith x (\<phi> a)) c w) \<and> \<I>(\<lambda>x. pv w x \<and> \<not> (\<forall>xa. (\<^bold>\<not>InterferesWith xa (\<phi> a)) c x))"
epath_shift: fixes srs :: "('sequent \<times> 'rule) list"   and steps :: "('sequent \<times> 'rule) stream" assumes "epath (srs @- steps)" shows "epath steps"
assms: fixes s :: "'state"   and s1 :: "'state"   and vl :: "'value list"   and vl1 :: "'value list" assumes "reachNT s"   and "reach s1"   and "\<Delta> s vl s1 vl1" shows "vl \<noteq> [] \<and> exit s (hd vl) \<or> iaction (disjAll \<Delta>s) s vl s1 vl1 \<or> (vl \<noteq> [] \<or> vl1 = []) \<and> reaction (disjAll \<Delta>s) s vl s1 vl1"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> parts H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> parts H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P X"   and "\<And>X Y. \<lbrakk>\<lbrace>X, Y\<rbrace> \<in> parts H; P \<lbrace>X, Y\<rbrace>\<rbrakk> \<Longrightarrow> P Y"   and "\<And>K X. \<lbrakk>Crypt K X \<in> parts H; P (Crypt K X)\<rbrakk> \<Longrightarrow> P X"   and "\<And>x y. \<lbrakk>Auth_Data x y \<in> parts H; P (Auth_Data x y)\<rbrakk> \<Longrightarrow> P (Pri_AgrK x)"   and "\<And>x y. \<lbrakk>Auth_Data x y \<in> parts H; P (Auth_Data x y)\<rbrakk> \<Longrightarrow> P (Pri_AgrK y)" shows "P x"
card_rcosets_equal: fixes G :: "('c, 'd) monoid_scheme"   and R :: "'c set"   and H :: "'c set" assumes "Group.group G"   and "R \<in> rcosets\<^bsub>G\<^esub> H"   and "H \<subseteq> carrier G" shows "card H = card R"
sum_mono_subseteq: fixes K :: "'b set"   and f :: "'b \<Rightarrow> 'c zmultiset"   and g :: "'b \<Rightarrow> 'c zmultiset" assumes "\<And>i. i \<in> K \<Longrightarrow> f i \<subseteq>#\<^sub>z g i" shows "sum f K \<subseteq>#\<^sub>z sum g K"
term_variants_pred_iff_in_term_variants: fixes P :: "'e \<Rightarrow> 'e list"   and t :: "('e, 'f) Term.term"   and s :: "('e, 'f) Term.term" shows "term_variants_pred P t s = (s \<in> set (term_variants P t))"
cancel_right: fixes ys :: "'b list"   and xs :: "'b list"   and zs :: "'b list" shows "(ys \<cdot> xs = zs \<cdot> xs) = (ys = zs)"
lsteps_smaller_set: fixes m :: "nat"   and l :: "nat"   and xs :: "'a list"   and f :: "'a \<Rightarrow> nat"   and u :: "nat"   and v :: "nat" assumes "m \<le> l" shows "lsteps xs f l u v = lsteps (filter (\<lambda>x. m \<le> f x) xs) f l u v"
simps: fixes a :: "('fenv_inca, 'henv, ('fun, 'dyn) frame) state" shows "Sinca.final a = (\<exists>F f fd pc H \<Sigma>. a = State F H [Frame f pc \<Sigma>] \<and> Finca_get F f = Some fd \<and> pc = length (body fd))"
\<phi>_def: shows "\<phi> = fut_derivs i \<psi>"
set_subset_Univ: fixes a :: "'s" assumes "ide a" shows "local.set a \<subseteq> Univ"
itop_def: fixes A :: "'b set" shows "itop A = topo (\<Union>s\<in>A\<^sup>\<star>. {infsuff A s})"
simps: fixes r :: "le_rel"   and p :: "linear_poly"   and c :: "rat" shows "of_rat_lec (Le_Constraint r p c) = Le_Constraint r p (real_of_rat c)"
cancel_separate_eq: fixes x :: "'c \<times> 'c \<Rightarrow> 'd"   and y :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_idempotent_semiring.coreflexive (x \<odot> y)" shows "x\<^sup>\<odot> \<odot> y\<^sup>\<odot> = x\<^sup>\<odot> \<oplus> y\<^sup>\<odot>"
OpposedThinChamberComplexFoldings_def: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and g :: "'b \<Rightarrow> 'b"   and C0 :: "'b set" shows "OpposedThinChamberComplexFoldings X f g C0 \<equiv> (ThinChamberComplex X \<and> ChamberComplexFolding X f) \<and> ChamberComplexFolding X g \<and> OpposedThinChamberComplexFoldings_axioms X f g C0"
det_state_cover_d_reachable: fixes M :: "('e, 'f, 'g) FSM"   and V :: "'e list set"   and v :: "'e list" assumes "is_det_state_cover M V"   and "v \<in> V" obtains q :: "'g" where "d_reaches M (initial M) v q"
test_bit_split: fixes c :: "'d word"   and a :: "'b word"   and b :: "'c word" assumes "word_split c = (a, b)" shows "(\<forall>n. b !! n = (n < size b \<and> c !! n)) \<and> (\<forall>m. a !! m = (m < size a \<and> c !! (m + size b)))"
subset_emptyI: fixes A :: "'d set" assumes "\<And>x. x \<in> A \<Longrightarrow> False" shows "A \<subseteq> {}"
hyps_3: fixes c :: "('a, 'b, 'c) com" shows "\<Gamma>\<turnstile> (subst_redex c r'', s'') \<rightarrow>\<^sup>* (subst_redex c r', s')"
mk_conn_orderB: fixes H1 :: "'b hrel \<Rightarrow> 'b hrel"   and \<H>\<^sub>1 :: "'a hrel \<Rightarrow> 'b hrel"   and \<H>\<^sub>2 :: "'b hrel \<Rightarrow> 'a hrel"   and H2 :: "'a hrel \<Rightarrow> 'a hrel" shows "\<Y>\<^bsub>H1 \<Leftarrow>\<langle>\<H>\<^sub>1,\<H>\<^sub>2\<rangle>\<Rightarrow> H2\<^esub> = utp_order H2"
prime_elem_iff_irreducible: 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" assumes "class.factorial_semiring divide plus minus zero times one unit_factor normalize" shows "comm_semiring_1.prime_elem times one zero x = comm_semiring_1.irreducible times one zero x"
one_final_but_def: fixes g :: "graph"   and E :: "(nat \<times> nat) set" shows "one_final_but g E \<equiv> \<forall>f\<in>\<F> g. \<not> final f \<longrightarrow> (\<forall>(a, b)\<in>\<E> f - E. (b, a) \<in> E \<or> (\<exists>f'\<in>\<F> g. final f' \<and> (b, a) \<in> \<E> f'))"
assms_3: shows "asc_fault_domain M2 M1 m"
map_indets_def: fixes f :: "'c \<Rightarrow> 'e" shows "map_indets f = poly_subst (\<lambda>x. monomial (1::'d) (monomial 1 (f x)))"
intro: fixes hom :: "'a \<Rightarrow> 'b" assumes "monoid_add_hom_0 hom" shows "comm_monoid_add_hom_0 hom"
fin_chain_def: fixes f :: "nat \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and Q :: "'a set" shows "[f[x .. y]Q] \<equiv> short_ch Q \<and> x \<in> Q \<and> y \<in> Q \<and> x \<noteq> y \<or> (\<exists>z\<in>Q. [f[x .. z ..  y]Q])"
poly_coeff_sum_def: fixes p :: "('d monom \<times> 'c) list" shows "poly_coeff_sum p \<equiv> \<Sum>mc\<leftarrow>p. max (0::'c) (snd mc)"
invar_delete: fixes t :: "'b trie_map"   and xs :: "'b list" assumes "invar t" shows "invar (Trie_Map.delete xs t)"
degree_of_X_minus: fixes a :: "'a" assumes "a \<in> carrier R"   and "\<one> \<noteq> \<zero>" shows "deg R (X_poly_minus R a) = 1"
false: assumes "do_if v e1 e2 = Some e2" shows "thesis"
cross_alt_coprime_0: fixes p :: "real poly"   and q :: "real poly"   and a :: "real"   and b :: "real" assumes "coprime p q"   and "p = 0 \<or> q = 0" shows "cross_alt p q a b = 0"
abs_eq: shows "Z' = abs_var (BPS (bsingleton (True, undefined)))"
subset_in_close: fixes S :: "('a, 'b) form set"   and S' :: "('a, 'b) form set"   and x :: "('a, 'b) form set"   and C :: "('a, 'b) form set set" assumes "S \<subseteq> S'" shows "S' \<union> x \<in> C \<longrightarrow> S \<union> x \<in> close C"
distinct_9: fixes x61 :: "'b ltln"   and x62 :: "'b ltln" shows "true\<^sub>n \<noteq> x61 or\<^sub>n x62"
path_ex_obs_intra: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node"   and S :: "'node set" assumes "n -as\<rightarrow>\<^sub>\<iota>* n'"   and "n' \<in> S" obtains m :: "'node" where "m \<in> obs_intra n S"
linear_simps_1: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'d"   and b :: "'d" assumes "bounded_linear f" shows "f (a + b) = f a + f b"
graph_extD2: fixes H :: "'c set"   and h :: "'c \<Rightarrow> real"   and H' :: "'c set"   and h' :: "'c \<Rightarrow> real" assumes "graph H h \<subseteq> graph H' h'" shows "H \<subseteq> H'"
right_idem: fixes a :: "'a"   and b :: "'a" shows "(a \<sqinter> b) \<sqinter> b = a \<sqinter> b"
simps_71: fixes x1 :: "'e" shows "set2_Kform (Kprop x1) = {x1}"
assms_1: assumes "f \<in> space\<^sub>N N1" shows "Norm N2 f \<le> C * Norm N1 f"