hcju/isabelleps huggingface dataset:
intro\<^sub>1: fixes \<Gamma> :: "'Var \<Rightarrow> Sec option"   and lc :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and lc' :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "(case lc of (x, xa) \<Rightarrow> (case x of (x, xa) \<Rightarrow> \<lambda>xaa ((xaaa, xaaaa), xaaaaa). \<langle>x, xa, xaa\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>\<^esub> \<langle>xaaa, xaaaa, xaaaaa\<rangle>) xa) lc'" shows "lc \<R>\<^sup>u\<^bsub>\<Gamma>\<^esub> lc'"
nth_solution_sound: fixes n :: "nat" shows "solution (nth_solution n)"
0: fixes b :: "'a local.term" assumes "Ide a \<and> Nml a"   and "Ide b \<and> Nml b"   and "Src a = Trg b" shows "Can (a \<^bold>\<Down> b) \<and> a \<^bold>\<Down> b \<in> VHom (a \<^bold>\<star> b) \<^bold>\<lfloor>a \<^bold>\<star> b\<^bold>\<rfloor>"
app_iso_def: fixes hom :: "('e, 'f, 'c, 'd) digraph_isomorphism"   and G :: "('e, 'f) pre_digraph" shows "app_iso hom G \<equiv> \<lparr>pre_digraph.verts = iso_verts hom ` pre_digraph.verts G, arcs = iso_arcs hom ` pre_digraph.arcs G, tail = iso_tail hom, head = iso_head hom\<rparr>"
g': shows "g' \<in> UNIV \<rightarrow> {- 1<..<1}"
nz: shows "blinfun_apply (Ds (flow0 x t)) (f (flow0 x t)) \<noteq> 0"
ordered_subdag: fixes t :: "dag"   and var :: "ref \<Rightarrow> nat"   and not :: "dag" assumes "ordered t var"   and "not \<le> t" shows "ordered not var"
llist_all2_LNil_LCons: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and y21 :: "'d"   and y22 :: "'d llist" shows "\<not> llist_all2 R LNil (LCons y21 y22)"
simps_3: fixes xs :: "cell list" shows "bl_bin (Oc # xs) = Suc (2 * bl_bin xs)"
\<Delta>'''_sel_1: fixes \<Delta> :: "('a, 'b) network_scheme"   and x :: "'a"   and y :: "'a" assumes "countable_network \<Delta>" shows "edge (countable_network.\<Delta>''' \<Delta>) x y = (edge \<Delta> x y \<and> 0 < capacity \<Delta> (x, y) \<and> y \<noteq> source \<Delta> \<and> x \<noteq> sink \<Delta>)"
deleteMin_mset: fixes q :: "('c, 'd) BinomialTree list" assumes "queue_invar q"   and "q \<noteq> []" shows "queue_to_multiset (deleteMin q) = queue_to_multiset q - {#findMin q#}"
elims: fixes x :: "('d, 'f) Term.term"   and xa :: "'f \<Rightarrow> ('d, 'e) Term.term"   and y :: "('d, 'e) Term.term" assumes "x \<cdot> xa = y" obtains xb :: "'f" and \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term" where "x = Var xb" and "xa = \<sigma>__" and "y = \<sigma>__ xb"   | f :: "'d" and ss :: "('d, 'f) Term.term list" and \<sigma> :: "'f \<Rightarrow> ('d, 'e) Term.term" where "x = Fun f ss" and "xa = \<sigma>" and "y = Fun f (ss \<cdot>\<^sub>l\<^sub>i\<^sub>s\<^sub>t \<sigma>)"
section_and_retraction_eq_homeomorphic_map: fixes X :: "'c topology"   and Y :: "'d topology"   and f :: "'c \<Rightarrow> 'd" shows "(section_map X Y f \<and> retraction_map X Y f) = homeomorphic_map X Y f"
dickson_grading_minus: fixes d :: "'a \<Rightarrow> nat"   and s :: "'a"   and t :: "'a" assumes "dickson_grading d"   and "s adds t" shows "d (t - s) \<le> d t"
measurable_distr_eq2: fixes Mg' :: "'b measure"   and Mg :: "'d measure"   and Ng :: "'c measure"   and g :: "'d \<Rightarrow> 'c" shows "Mg' \<rightarrow>\<^sub>M distr Mg Ng g = Mg' \<rightarrow>\<^sub>M Ng"
jordan_nf_smult: fixes A :: "'a mat"   and n_as :: "(nat \<times> 'a) list"   and k :: "'a" assumes "jordan_nf A n_as"   and "k \<noteq> (0::'a)" shows "jordan_nf (k \<cdot>\<^sub>m A) (map (\<lambda>(n, a). (n, k * a)) n_as)"
cases: fixes a1 :: "'h"   and a2 :: "'i"   and r :: "('i \<times> 'h) set" assumes "(a1, a2) \<in> r\<inverse>" obtains a :: "'i" and b :: "'h" where "a1 = b" and "a2 = a" and "(a, b) \<in> r"
simps: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'b stream" assumes "class.order less_eq less" shows "order.sdescending less_eq a = (\<exists>a b w. a = a ## b ## w \<and> less_eq b a \<and> order.sdescending less_eq (b ## w))"
opt_bst2_is_optimal: fixes w :: "int \<Rightarrow> int \<Rightarrow> nat"   and i :: "int"   and j :: "int" assumes "Optimal_BST2 w" shows "wpl.wpl w i j (Optimal_BST2.opt_bst2 w i j) = Optimal_BST.min_wpl w i j"
smult_mult1: fixes a :: "'c"   and A :: "('c, 'd) sq_matrix"   and B :: "('c, 'd) sq_matrix" shows "a *\<^sub>S (A * B) = a *\<^sub>S A * B"
\<phi>_def: shows "\<phi> \<equiv> \<phi>.map"
Inter_iff: fixes A :: "'c"   and C :: "'c set set" shows "(A \<in> \<Inter> C) = (\<forall>X\<in>C. A \<in> X)"
if_SE_def: fixes c :: "'a \<Rightarrow> bool"   and E :: "'a \<Rightarrow> ('b \<times> 'a) option"   and F :: "'a \<Rightarrow> ('b \<times> 'a) option" shows "_if_SE c E F = (\<lambda>\<sigma>. if c \<sigma> then E \<sigma> else F \<sigma>)"
simps_3: fixes P :: "'a pred"   and xq :: "'a seq"   and x :: "'a" shows "member (Join P xq) x = (eval P x \<or> member xq x)"
der: fixes x :: "real" assumes "b \<le> x" shows "\<exists>y. (f has_real_derivative y) (at x) \<and> y < 0"
respects_hcomp: fixes f :: "'c"   and g :: "'c" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C g = trg\<^sub>C f" shows "(\<sigma>\<^sub>0 (trg\<^sub>C g) \<star>\<^sub>D \<Phi>\<^sub>F (g, f)) \<cdot>\<^sub>D \<a>\<^sub>D[\<sigma>\<^sub>0 (trg\<^sub>C g), F g, F f] \<cdot>\<^sub>D (\<sigma>\<^sub>1 g \<star>\<^sub>D F f) \<cdot>\<^sub>D D.inv \<a>\<^sub>D[G g, \<sigma>\<^sub>0 (src\<^sub>C g), F f] \<cdot>\<^sub>D (G g \<star>\<^sub>D \<sigma>\<^sub>1 f) \<cdot>\<^sub>D \<a>\<^sub>D[G g, G f, \<sigma>\<^sub>0 (src\<^sub>C f)] = \<sigma>\<^sub>1 (g \<star>\<^sub>C f) \<cdot>\<^sub>D (\<Phi>\<^sub>G (g, f) \<star>\<^sub>D \<sigma>\<^sub>0 (src\<^sub>C f))"
rel_I1: fixes s :: "'b"   and U :: "'a set"   and x :: "'a set \<times> 'b" assumes "s \<in> \<FF> U"   and "x \<in> \<lfloor>U, s\<rfloor>"   and "U \<in> I" shows "(U, s) \<sim> x"
bind_spmf_cong_simp: fixes p :: "'d spmf"   and q :: "'d spmf"   and f :: "'d \<Rightarrow> 'e spmf"   and g :: "'d \<Rightarrow> 'e spmf" assumes "p = q"   and "\<And>x. x \<in> set_spmf q =simp=> f x = g x" shows "p \<bind> f = q \<bind> g"
Dom_def: fixes problem :: "(char list \<times> nat option \<times> char list list) list \<times> nat list \<times> (nat \<times> nat) list \<times> (char list \<times> (nat \<times> nat) list \<times> ((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list \<times> nat) list" shows "ast_problem.Dom problem = {0..<ast_problem.numVars problem}"
simps_2: fixes e :: "'a"   and a :: "'b"   and t :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree" shows "SkewBinomialHeapStruc.insert e a [t] = [SkewBinomialHeapStruc.Node e a 0 [], t]"
mndet_distrib_unit: fixes A :: "'a set"   and a :: "'a"   and P :: "'a \<Rightarrow> 'a process" assumes "A - {a} \<noteq> {}" shows "mndet (insert a A) P = (a \<rightarrow> P a |-| mndet (A - {a}) P)"
effective_scalar_product_distributivity: fixes v1 :: "'a list"   and v2 :: "'a list"   and w1 :: "'a list"   and w2 :: "'a list" assumes "length v1 = length v2"   and "length w1 = length w2" shows "scalar_product v1 v2 * scalar_product w1 w2 = scalar_product (vec_vec_Tensor v1 w1) (vec_vec_Tensor v2 w2)"
assms_2: shows "d (pp_of_term u) \<le> m"
quasi_inverts_components_2: fixes a :: "'a" assumes "C.obj a" shows "D.isomorphic (\<tau>\<^sub>0 a \<star>\<^sub>D map\<^sub>0 a) (G.map\<^sub>0 a)"
assms_2: shows "((\<sigma>, s), a, \<sigma>', s') \<in> T"
rotate_drop_take: fixes n :: "nat"   and xs :: "'c list" shows "rotate n xs = drop (n mod |xs|) xs @ take (n mod |xs|) xs"
is_aligned_mask_offset_unat: fixes x :: "'b word"   and sz :: "nat"   and off :: "'b word" assumes "is_aligned x sz"   and "off \<le> mask sz" shows "unat x + unat off < 2 ^ LENGTH('b)"
ET_HADelta: fixes TS :: "('d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd) set"   and ST :: "('d, 'e, 'f) status"   and t :: "'d \<times> (('d, 'e) expr \<times> ('f data \<Rightarrow> bool) \<times> 'e set \<times> 'f pupdate) \<times> 'd" assumes "TS \<subseteq> ET ST"   and "t \<in> TS" shows "t \<in> HADelta (HA ST)"
inverse_composition_commute: fixes x :: "'a"   and y :: "'a" assumes "invertible x"   and "invertible y"   and "x \<in> G"   and "y \<in> G" shows "inverse (x \<cdot> y) = inverse y \<cdot> inverse x"
pw_rwof_init_2: fixes m0 :: "'a nres"   and cond :: "'a \<Rightarrow> bool"   and step :: "'a \<Rightarrow> 'a nres" assumes "nofail (m0 \<bind> WHILE cond step)" shows "nofail m0"
qfa_cons: shows "qfa (x # xs) = do_act x (qfa xs)"
mult_of_nat_commute: fixes one :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and x :: "nat"   and y :: "'k" assumes "class.semiring_1 one times plus zero" shows "times (semiring_1.of_nat one plus zero x) y = times y (semiring_1.of_nat one plus zero x)"
equal_odlist_def: fixes A :: "'a odlist"   and B :: "'a odlist" shows "equal_class.equal A B = odlist_equal (toList A) (toList B)"
simps_4: fixes x1 :: "rat"   and x21 :: "rat"   and x22 :: "rat" shows "Rat_Info x1 \<noteq> Sqrt_Info x21 x22"
compare_LT_trans: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "compare\<cdot>x\<cdot>y = LT"   and "compare\<cdot>y\<cdot>z = LT" shows "compare\<cdot>x\<cdot>z = LT"
sym_sum_inter_num_sq: fixes b1 :: "'a set" assumes "b1 \<in># \<B>" shows "(\<Sum>bl\<in>#remove1_mset b1 \<B>. (b1 |\<inter>| bl)\<^sup>2) = \<Lambda>\<^sup>2 * (\<v> - 1)"
i2: shows "\<forall>d\<in>D. bal (dblock s p) < mbal (disk s d q)"
inject: fixes x :: "char list"   and ya :: "char list" shows "(Iface x = Iface ya) = (x = ya)"
induct: fixes P :: "Core.term \<Rightarrow> nat \<Rightarrow> Core.term list \<Rightarrow> bool"   and a0 :: "Core.term"   and a1 :: "nat"   and a2 :: "Core.term list" assumes "\<And>n lev args. P (Bv n) lev args"   and "\<And>T body lev args. P body (lev + 1) args \<Longrightarrow> P (Abs T body) lev args"   and "\<And>f t lev args. \<lbrakk>P f lev args; P t lev args\<rbrakk> \<Longrightarrow> P (f $ t) lev args"   and "\<And>v va uu_ uv_. P (Ct v va) uu_ uv_"   and "\<And>v va uu_ uv_. P (Fv v va) uu_ uv_" shows "P a0 a1 a2"
simps_11: fixes x31 :: "'b"   and x32 :: "('a, 'b) form list"   and x21 :: "'a"   and x22 :: "('a, 'b) form list" shows "Modal x31 x32 \<noteq> Compound x21 x22"
assoc: fixes a :: "'a"   and b :: "'a"   and c :: "'a" shows "a + b + c = a + (b + c)"
abs_eq: shows "0 = ZHMSet {#}\<^sub>z"
Id_refine: fixes S :: "'a nres" shows "S \<le> \<Down> Id S"
order_iff_strict: fixes a :: "'d word"   and b :: "'d word" shows "(a \<le>s b) = (a <s b \<or> a = b)"
subprob_emeasure_le_1: fixes p :: "'c spmf"   and X :: "'c set" shows "emeasure (measure_spmf p) X \<le> 1"
intY1_is_cl: shows "\<lparr>pset = intY1, order = induced intY1 r\<rparr> \<in> CompleteLattice"
strict_iff_order: fixes b :: "'c word"   and a :: "'c word" shows "(b <s a) = (b \<le>s a \<and> a \<noteq> b)"
inorder_upd: fixes t :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and x :: "'c"   and y :: "'d" assumes "sorted1 (Tree2.inorder t)" shows "Tree2.inorder (upd x y t) = upd_list x y (Tree2.inorder t)"
bind_sample2: fixes m :: "'m"   and p :: "'p pmf"   and f :: "'a option \<Rightarrow> 'p \<Rightarrow> 'm" shows "bind m (\<lambda>y. sample p (f y)) = sample p (\<lambda>x. bind m (\<lambda>y. f y x))"
weight_w0: fixes t :: "('f, 'b) Term.term" shows "w0 \<le> weight t"
TC_soundness: fixes G :: "('d, 'e) form list"   and e :: "nat \<Rightarrow> 'f"   and f :: "'d \<Rightarrow> 'f list \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'f list \<Rightarrow> bool" assumes "\<stileturn> G" shows "\<exists>p\<in>set G. \<not> eval e f g p"
trans_init_seg_of: fixes r :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and t :: "('b \<times> 'b) set" assumes "r initial_segment_of s"   and "s initial_segment_of t" shows "r initial_segment_of t"
oth_class_taut_8_b: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>"   and \<chi> :: "\<o>" shows "[\<phi> \<^bold>\<rightarrow> (\<psi> \<^bold>\<rightarrow> \<chi>) \<^bold>\<rightarrow> (\<phi> \<^bold>& \<psi> \<^bold>\<rightarrow> \<chi>) in v]"
quote_perm_def: fixes p :: "perm"   and Vs :: "name set" shows "quote_perm p Vs \<equiv> atom ` (p \<bullet> Vs) \<sharp>* Vs \<and> - p = p \<and> finite Vs"
ac_Zp_factors_x_1: fixes x :: "nat \<Rightarrow> int" assumes "x \<in> carrier Zp"   and "x \<noteq> \<zero>" shows "x = [p] \<cdot> \<one> [^] nat (ord_Zp x) \<otimes> ac_Zp x"
set_child_nodes_get_tag_name: fixes ptr :: "(_) object_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" shows "\<forall>w\<in>Shadow_DOM.set_child_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_tag_name_locs ptr'. r h h')"
finite_same_card_bij: fixes A :: "'a set"   and B :: "'b set" assumes "finite A"   and "finite B"   and "card A = card B" shows "\<exists>h. bij_betw h A B"
int_not_simps_3: shows "NOT (- 1) = 0"
insert: fixes A :: "'a set"   and x :: "'a"   and g :: "'a \<Rightarrow> 'b" assumes "finite A"   and "x \<notin> A" shows "sum g (insert x A) = g x + sum g A"
invar_del_min: fixes ts :: "'b tree list" assumes "ts \<noteq> []"   and "invar ts" shows "invar (del_min ts)"
D: shows "s\<^sub>0 \<Turnstile> u"
empty_imp_not_in: fixes x :: "'b" shows "x \<notin> {}"
l_get_dom_component_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_axioms: shows "l_get_dom_component_create_document\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M DocumentClass.known_ptr heap_is_wellformed parent_child_rel DocumentClass.type_wf DocumentClass.known_ptrs to_tree_order get_parent get_parent_locs get_child_nodes get_child_nodes_locs get_dom_component is_strongly_dom_component_safe is_weakly_dom_component_safe get_root_node get_root_node_locs get_ancestors get_ancestors_locs get_element_by_id get_elements_by_tag_name create_document"
rb_aux_inv2_D1: fixes args :: "('t \<Rightarrow>\<^sub>0 'b) list \<times> 't list \<times> (('t \<Rightarrow>\<^sub>0 'b) \<times> ('t \<Rightarrow>\<^sub>0 'b) + nat) list" assumes "rb_aux_inv2 args" shows "rb_aux_inv args"
rsemimodule1: fixes s :: "'b"   and t :: "'b"   and a :: "'a" shows "\<lambda> (s \<oplus> t, a) = \<lambda> (s, a) \<oplus> \<lambda> (t, a)"
before_def: fixes vs :: "'a list"   and ram1 :: "'a"   and ram2 :: "'a" shows "before vs ram1 ram2 \<equiv> \<exists>a b c. vs = a @ ram1 # b @ ram2 # c"
lim14: shows "((\<lambda>x. 3 * x / (1 + 3 * x\<^sup>2)) \<longlongrightarrow> 0) at_infinity"
assms: shows "s' = snd (fst (logical_instr instr s)) \<and> get_S (cpu_reg_val PSR s) = 0"
shows_mat_def: fixes A :: "'b mat" shows "shows_mat A \<equiv> shows (mat_to_list A)"
assms: shows "set \<Phi> \<subseteq> set \<Phi>'"
v: shows "(\<lambda>n. Bcontfun (u n)) \<longlonglongrightarrow> v"
card_image: fixes f :: "'g \<Rightarrow> 'h"   and A :: "'g set" assumes "inj_on f A" shows "card (f ` A) = card A"
zero_less_mult_pos: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.linordered_semiring_strict plus zero minus less_eq less times"   and "less zero (times a b)"   and "less zero a" shows "less zero b"
IH_prem2: fixes f :: "'a match_expr \<Rightarrow> 'a match_expr option"   and m :: "'a match_expr" assumes "f \<in> set fs"   and "normalized_nnf_match m"   and "f m = None" shows "\<not> matches \<gamma> m a p"
sound_modes_invariant: fixes cms :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem :: "'Var \<Rightarrow> 'Val"   and cms' :: "('Com \<times> (Mode \<Rightarrow> 'Var set)) list"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "sound_mode_use (cms, mem)"   and "(cms, mem) \<rightarrow> (cms', mem')" shows "sound_mode_use (cms', mem')"
solves_ode_domainD: fixes y :: "real \<Rightarrow> 'b"   and f :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and T :: "real set"   and X :: "'b set"   and t :: "real" assumes "(y solves_ode f) T X"   and "t \<in> T" shows "y t \<in> X"
set_eq: fixes x1 :: "real" shows "{x \<in> space (lborel_f n). x(n := x1) \<in> A} = {f \<in> space (lborel_f n). insertion f (p_fix1 x1) = 0}"
complex_add_cnj: fixes z :: "complex" shows "z + cnj z = complex_of_real (2 * Re z)"
u_1: fixes n :: "nat" shows "esssup M f < u n"
item_end_of_inc_item: fixes x :: "item"   and k :: "nat" shows "item_end (inc_item x k) = k"
hyps: fixes \<D> :: "'a set set" assumes "card \<D> < card \<D>'__"   and "\<D> division_of cbox a b" shows "{k \<in> \<D>. content k \<noteq> 0} division_of cbox a b"
Smith_2x2_JNF_eucl_def: fixes A :: "'b mat" shows "Smith_2x2_JNF_eucl A = (let (P, D, Q) = diagonalize_2x2_JNF A; (P', S, Q') = diagonal_to_Smith_PQ_JNF D euclid_ext2 in (P' * P, S, Q * Q'))"
trm_simps_1: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b" assumes "category_with_terminal_object C"   and "partial_magma.arr C f" shows "partial_magma.arr C (category_with_terminal_object.trm C f)"
less_imp_le: fixes less_eq :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and less :: "'f \<Rightarrow> 'f \<Rightarrow> bool"   and x :: "'f"   and y :: "'f" assumes "class.preorder less_eq less"   and "less x y" shows "less_eq x y"
\<eta>': shows "\<eta>' = D.inv (\<Phi> (g, f)) \<cdot>\<^sub>D F \<eta> \<cdot>\<^sub>D unit (src\<^sub>C f)"
assms_1: shows "P' \<in> carrier_mat m m \<and> A' \<in> carrier_mat m n \<and> Q' \<in> carrier_mat n n"
less_asym: fixes x :: "'a"   and y :: "'a"   and P :: "bool" assumes "x < y"   and "\<not> P \<Longrightarrow> y < x" shows "P"
simps_1: fixes A :: "bool"   and uu :: "bool"   and uv :: "'arr list" shows "Dom (MkArr A uu uv) = A"
K_cfg_rept_action: fixes cfg :: "('s \<times> ('c \<Rightarrow> real) set) cfg"   and s :: "'s \<times> ('c \<Rightarrow> real)"   and cfg' :: "('s \<times> ('c \<Rightarrow> real) set) cfg" assumes "cfg \<in> R_G.valid_cfg"   and "abss s = state cfg"   and "cfg' \<in> set_pmf (K_cfg cfg)" shows "abss (THE s'. s' \<in> set_pmf (rept s (action cfg)) \<and> abss s' = state cfg') = state cfg'"
simps_18: fixes f1 :: "ipc_direction_t \<Rightarrow> ipc_stage_t \<Rightarrow> thread_id_t \<Rightarrow> page_t \<Rightarrow> 'a"   and f2 :: "ev_wait_stage_t \<Rightarrow> ev_consume_t \<Rightarrow> 'a"   and f3 :: "ev_signal_stage_t \<Rightarrow> thread_id_t \<Rightarrow> 'a"   and f4 :: "'a"   and x31 :: "ev_signal_stage_t"   and x32 :: "thread_id_t" shows "(case SK_EV_SIGNAL x31 x32 of SK_IPC x xa xb xc \<Rightarrow> f1 x xa xb xc | SK_EV_WAIT x xa \<Rightarrow> f2 x xa | SK_EV_SIGNAL x xa \<Rightarrow> f3 x xa | NONE \<Rightarrow> f4) = f3 x31 x32"
permutations_of_set_def: fixes A :: "'b set" shows "permutations_of_set A = {xs. set xs = A \<and> distinct xs}"
Notes_TTP_imp_Gets: fixes A :: "agent"   and B :: "agent"   and L :: "nat"   and K :: "nat"   and con_K :: "msg"   and evs :: "event list"   and sub_K :: "msg" assumes "Notes TTP \<lbrace>Number f_con, Agent A, Agent B, Nonce L, Key K, con_K\<rbrace> \<in> set evs"   and "sub_K = Crypt (priEK A) \<lbrace>Number f_sub, Agent B, Nonce L, Key K\<rbrace>"   and "evs \<in> zg" shows "Gets TTP \<lbrace>Number f_sub, Agent B, Nonce L, Key K, sub_K\<rbrace> \<in> set evs"
s2_def: shows "s2 = the_inv_into (\<Union>(f, g)\<in>fundfoldpairs. {Abs_induced_automorph f g}) fundantivertex (\<phi> v2)"
I_3: shows "subst_domain I = \<V> \<and> ground (subst_range I)"
qs: shows "qs = map (abs_int_poly \<circ> fst) (snd (c, qis))"
keys_higherPoly\<^sub>0: fixes b :: "nat"   and i :: "nat"   and mp :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "keys (higherPoly\<^sub>0 b i mp) = lowerPowers b i ` keys mp"
mult_divide_mult_cancel_right: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and c :: "'a"   and a :: "'a"   and b :: "'a" assumes "class.field divide inverse times one plus zero minus uminus"   and "c \<noteq> zero" shows "divide (times a c) (times b c) = divide a b"
assms_3: shows "F \<subseteq> dgrad_p_set d m"
Sup_dres_def: fixes S :: "'b dres set" shows "Sup S \<equiv> if S \<subseteq> {dSUCCEEDi} then dSUCCEEDi else if dFAILi \<in> S then dFAILi else if \<exists>a b. a \<noteq> b \<and> dRETURN a \<in> S \<and> dRETURN b \<in> S then dFAILi else dRETURN (THE x. dRETURN x \<in> S)"
move_filter: shows "move = filter (\<lambda>(t, X). left I \<le> nt - t) (linearize data_prev)"
Abs_errorable_inverse: fixes y :: "'b option" assumes "y \<in> UNIV" shows "Rep_errorable (Abs_errorable y) = y"
var_order_rel_antisym: fixes y :: "char list" shows "(y, y) \<notin> var_order_rel"
stream_pred_snth: fixes P :: "'a \<Rightarrow> bool"   and w :: "'a stream" shows "pred_stream P w = (\<forall>i. P (w !! i))"
rep_nc: shows "\<forall>pt i. pt \<notin> set_of pret \<or> n \<le> i \<and> pt \<in> set (ll ! i) \<and> i < var p + 1 \<longrightarrow> rep pt = repb pt"
assms_1: shows "integrable (lebesgue_on {a..b}) f"
fs_int_d_Suc: fixes k :: "nat" assumes "k < length fs" shows "rat_of_int (fs.d fs (Suc k)) = \<parallel>fs.gs.gso k\<parallel>\<^sup>2 * rat_of_int (fs.d fs k)"
exhaust: fixes y :: "[:'r:]" obtains "y = \<bottom>"   | "y = [::]"   | a :: "'r" and slist :: "[:'r:]" where "y = a :# slist" and "a \<noteq> \<bottom>" and "slist \<noteq> \<bottom>"
dup_naturality: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>" shows "\<d>[b] \<cdot> f = (f \<otimes> f) \<cdot> \<d>[a]"
\<gamma>_st_def: fixes \<gamma> :: "'b \<Rightarrow> 'a set"   and F :: "'b Abs_State.st" shows "\<gamma>_st \<gamma> F = {f. \<forall>x. f x \<in> \<gamma> (lookup F x)}"
gR_rel_def: fixes x :: "'c \<times> 'd"   and y :: "'c \<times> 'd" shows "ps_prod.gR_rel x y = (\<exists>z. ps_prod_D x z \<and> ps_prod_times x z = y)"
empty1_def: shows "empty1 \<equiv> RETURN (0::'b, 0::'c, [])"
LstSeq_imp_Seq_succ: fixes s :: "hf"   and k :: "hf"   and y :: "hf" assumes "LstSeq s k y" shows "Seq s (succ k)"
representant_exists: fixes A :: "'a set" assumes "A \<in> Partition" shows "\<exists>a\<in>S. a \<in> A \<and> A = Class a"
unique_expression5: fixes H :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'a"   and m :: "nat"   and t :: "nat \<Rightarrow> 'b" assumes "free_generator R M H"   and "f \<in> {j. j \<le> n} \<rightarrow> H"   and "inj_on f {j. j \<le> n}"   and "s \<in> {j. j \<le> n} \<rightarrow> carrier R"   and "g \<in> {j. j \<le> m} \<rightarrow> H"   and "inj_on g {j. j \<le> m}"   and "t \<in> {j. j \<le> m} \<rightarrow> carrier R"   and "l_comb R M n s f = l_comb R M m t g"   and "\<forall>j\<in>{j. j \<le> n}. s j \<noteq> \<zero>\<^bsub>R\<^esub>"   and "\<forall>k\<in>{j. j \<le> m}. t k \<noteq> \<zero>\<^bsub>R\<^esub>" shows "f ` {j. j \<le> n} \<subseteq> g ` {j. j \<le> m}"
pat_var: fixes p :: "pat"   and T :: "ty"   and \<Delta> :: "(name \<times> ty) list" assumes "\<turnstile> p : T \<Rightarrow> \<Delta>" shows "supp p = supp \<Delta>"
letter_eq_dec: fixes a :: "letter"   and b :: "letter" shows "a = b \<or> a \<noteq> b"
cl8: fixes x :: "'a"   and y :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> 1\<^sub>\<sigma> \<cdot> y = x \<cdot> 1\<^sub>\<pi> \<parallel> y"
aHom_def: fixes A :: "('d, 'f) aGroup_scheme"   and B :: "('e, 'g) aGroup_scheme" shows "aHom A B = {f \<in> carrier A \<rightarrow> carrier B. f \<in> extensional (carrier A) \<and> (\<forall>a\<in>carrier A. \<forall>b\<in>carrier A. f (a \<plusminus>\<^bsub>A\<^esub> b) = f a \<plusminus>\<^bsub>B\<^esub> f b)}"
replacement_theorem: fixes Us' :: "'a list"   and Us :: "'a list"   and Vs :: "'a list" assumes "local.independent K (Us' @ Us)"   and "local.independent K Vs"   and "Span K (Us' @ Us) \<subseteq> Span K Vs" shows "\<exists>Vs'. set Vs' \<subseteq> set Vs \<and> length Vs' = length Us' \<and> local.independent K (Vs' @ Us)"