hcju/isabelleps huggingface dataset:
StaticUpDynCastReds: fixes e :: "expr"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and a :: "nat"   and Cs :: "char list list"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and C :: "char list"   and Cs' :: "char list list"   and Ds :: "char list list" assumes "P,E \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>ref (a, Cs),s'\<rangle>"   and "P \<turnstile> Path last Cs to C unique"   and "P \<turnstile> Path last Cs to C via Cs' "   and "Ds = Cs @\<^sub>p Cs'" shows "P,E \<turnstile> \<langle>Cast C e,s\<rangle> \<rightarrow>* \<langle>ref (a, Ds),s'\<rangle>"
coeff_mpoly_of_poly2: fixes x :: "nat \<Rightarrow>\<^sub>0 nat"   and i :: "nat"   and p :: "'b poly" assumes "\<not> keys x \<subseteq> {i}" shows "MPoly_Type.coeff (mpoly_of_poly i p) x = (0::'b)"
label_incr_inj: fixes n :: "label"   and c :: "nat"   and n' :: "label" assumes "n \<oplus> c = n' \<oplus> c" shows "n = n'"
pth_4_1: fixes x :: "'d" shows "0 *\<^sub>R x \<equiv> 0::'d"
poly_power_n_eq: fixes n :: "nat"   and a :: "'b"   and x :: "'b" assumes "n \<noteq> 0" shows "(poly ([:- a, 1::'b:] ^ n) x = (0::'b)) = (x = a)"
encode_0: fixes e :: "nat" shows "local.encode e 0 = 0"
l1_implements_a0i_i: shows "implements med01iai a0i l1"
simps_1: fixes x :: "Preliminaries.var" shows "tpOf (Var x) = tpOfV x"
Some_eq_map_of_iff: fixes xys :: "('a \<times> 'b) list"   and y :: "'b"   and x :: "'a" assumes "distinct (map fst xys)" shows "(Some y = map_of xys x) = ((x, y) \<in> set xys)"
set_zip: fixes xs :: "'c list"   and ys :: "'d list" shows "set (zip xs ys) = {(xs ! i, ys ! i) |i. i < min (length xs) (length ys)}"
bpE_3: fixes U :: "'b set"   and w :: "'b \<Rightarrow> real"   and c :: "nat"   and P :: "'b set set" assumes "BinPacking_Complete U w c"   and "BinPacking_Complete.bp U w c P" shows "\<Union> P = U"
flip_def: fixes a :: "'a"   and b :: "'a" shows "(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
vertex_1_int_def: shows "vertex_1 \<equiv> 1"
generateI: fixes G :: "('c, 'd) monoid_scheme"   and E :: "'c set"   and H :: "'c set" assumes "Group.group G"   and "subgroup E G"   and "H \<subseteq> E"   and "\<And>K. \<lbrakk>subgroup K G; H \<subseteq> K\<rbrakk> \<Longrightarrow> E \<subseteq> K" shows "E = generate G H"
as_needed_asses_submap_exec_i: fixes v :: "'a"   and a :: "('a, 'b) fmap \<times> 'c"   and s :: "('a, 'b) fmap" assumes "v \<in> fmdom' (action_needed_asses a s)" shows "fmlookup (action_needed_asses a s) v = fmlookup s v \<and> fmlookup (action_needed_asses a s) v = fmlookup (fst a) v"
KS: fixes n :: "nat" shows "K n \<subseteq> S"
sterms_stermsl_heads: fixes ps :: "('a, 'b, 'c, 'd) seqp"   and \<Gamma> :: "'c \<Rightarrow> ('a, 'b, 'c, 'd) seqp"   and pn :: "'c" assumes "ps \<in> sterms \<Gamma> (\<Gamma> pn)"   and "wellformed \<Gamma>" shows "\<exists>pn. ps \<in> stermsl (\<Gamma> pn)"
assms_3: shows "is_zero_row_upt_k a (Suc k) A"
pmf_nonneg: fixes p :: "'g pmf"   and x :: "'g" shows "0 \<le> pmf p x"
assms: fixes i :: "'a" assumes "i \<in> I" shows "X i \<in> N \<rightarrow>\<^sub>M M i"
map_rw2: shows "map (sq_norm \<circ> row (sylvester_mat f g)) [degree g..<degree f + degree g] = replicate n \<parallel>g\<parallel>\<^sup>2"
continuous_on_scaleR: fixes s :: "'b set"   and f :: "'b \<Rightarrow> real"   and g :: "'b \<Rightarrow> 'c" assumes "continuous_on s f"   and "continuous_on s g" shows "continuous_on s (\<lambda>x. f x *\<^sub>R g x)"
antisymmetric: fixes x :: "'a"   and y :: "'a" assumes "le x y"   and "le y x" shows "x = y"
ActualImplI: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" assumes "[\<^bold>\<A>\<phi> \<^bold>\<rightarrow> \<^bold>\<A>\<psi> in v]" shows "[\<^bold>\<A>(\<phi> \<^bold>\<rightarrow> \<psi>) in v]"
v0_v1_notin_second_vertices_2: shows "v1 \<notin> second_vertices"
rerr_invalid_3: fixes crt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat) option"   and dests :: "nat \<Rightarrow> nat option"   and sip :: "nat" shows "rerr_invalid crt (Rerr dests sip) = (\<forall>rip\<in>dom dests. rip \<in> iD crt \<and> the (dests rip) = sqn crt rip)"
assms: shows "cscc_invar v0 D0 (l, p, D, pE)"
l11_36_aux1: 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 A' :: "'a"   and D :: "'a"   and E :: "'a"   and D' :: "'a"   and C :: "'a"   and F :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> B"   and "A' \<noteq> B"   and "D \<noteq> E"   and "D' \<noteq> E"   and "Bet A B A'"   and "Bet D E D'"   and "Tarski_neutral_dimensionless.LeA Bet Cong A B C D E F" shows "Tarski_neutral_dimensionless.LeA Bet Cong D' E F A' B C"
output_size_correct: fixes m :: "real mat convnet"   and inputs :: "real vec list" assumes "valid_net' m"   and "map dim_vec inputs = input_sizes m" shows "output_size' m = dim_vec (evaluate_net m inputs)"
abs_ast_operator_def: shows "abs_ast_operator \<equiv> \<lambda>(name, preconditions, effects, cost). \<lparr>precondition_of = preconditions, effect_of = map (\<lambda>(uu_, v, uu_, x). (v, x)) effects\<rparr>"
mult_eq_0_iff: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.semiring_no_zero_divisors plus zero times" shows "(times a b = zero) = (a = zero \<or> b = zero)"
measurable_count_space: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set" shows "f \<in> count_space A \<rightarrow>\<^sub>M count_space UNIV"
elementsEmptyIffTrailEmpty: fixes list :: "('b \<times> bool) list" shows "(elements list = []) = (list = [])"
bi_total_alt_def: fixes A :: "'c \<Rightarrow> 'd \<Rightarrow> bool" shows "bi_total A = (left_total A \<and> right_total A)"
subst_remove_var: fixes v :: "'a"   and s :: "('b, 'a) Term.term"   and t :: "('b, 'a) Term.term" assumes "v \<notin> fv s" shows "v \<notin> fv (t \<cdot> Var(v := s))"
inducts: fixes P :: "'a prog"   and U :: "ty"   and V :: "ty"   and T :: "ty" assumes "P \<turnstile> lub(U, V) = T" obtains "P \<turnstile> U \<le> T" and "P \<turnstile> V \<le> T" and "\<And>T'. \<lbrakk>P \<turnstile> U \<le> T'; P \<turnstile> V \<le> T'\<rbrakk> \<Longrightarrow> P \<turnstile> T \<le> T'"
simps_5: fixes v :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and va :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators"   and xs :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list" shows "wellformed_policy2Pr (v \<oplus> va # xs) = ((\<forall>c a b. c = DenyAllFromTo a b \<and> c \<in> set xs \<longrightarrow> dom (Cp (v \<oplus> va)) \<inter> dom (Cp c) = {}) \<and> wellformed_policy2Pr xs)"
start_method_def: fixes C :: "char list"   and M :: "char list" shows "start_method C M = (start_m, Static, [], Void, 1, 0, [Invokestatic C M 0, Return], [])"
diff_numeral_special_11: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and one :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "minus one (uminus one) = numeral.numeral one plus (num.Bit0 num.One)"
finite_valid_nodes: fixes prog :: "cmd" shows "finite {n. valid_node prog n}"
CASComp: fixes c\<^sub>e :: "expr"   and \<theta> :: "nat \<Rightarrow> nat option"   and D\<^sub>a :: "nat set"   and a :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat"   and s\<^sub>e :: "expr"   and A :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and L :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and R :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and W :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat set"   and t :: "nat" assumes "\<forall>sop. c\<^sub>e \<noteq> Tmp sop" shows "\<theta>\<turnstile> (CAS (Tmp (D\<^sub>a, a)) c\<^sub>e s\<^sub>e A L R W, t) \<rightarrow>\<^sub>p ((CAS (Tmp (D\<^sub>a, a)) (Tmp (eval_expr t c\<^sub>e)) s\<^sub>e A L R W, t + used_tmps c\<^sub>e), issue_expr t c\<^sub>e)"
and_wf_dfa: fixes M :: "nat bdd list \<times> bool list"   and n :: "nat"   and N :: "nat bdd list \<times> bool list" assumes "wf_dfa M n"   and "wf_dfa N n" shows "wf_dfa (and_dfa M N) n"
social_decision_scheme_axioms_def: fixes agents :: "'a set"   and alts :: "'b set"   and sds :: "('a \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'b pmf" shows "social_decision_scheme_axioms agents alts sds \<equiv> \<forall>R. pref_profile_wf agents alts R \<longrightarrow> sds R \<in> lotteries_on alts"
assms_2: shows "\<forall>(a, b)\<in>A. a \<noteq> b"
FOREACHoci_rule': fixes S :: "'c set"   and I :: "'c set \<Rightarrow> 'd \<Rightarrow> bool"   and \<sigma>0 :: "'d"   and c :: "'d \<Rightarrow> bool"   and R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'd nres"   and P :: "'d \<Rightarrow> bool" assumes "finite S"   and "I S \<sigma>0"   and "\<And>x it \<sigma>. \<lbrakk>c \<sigma>; x \<in> it; it \<subseteq> S; I it \<sigma>; \<forall>y\<in>it - {x}. R x y; \<forall>y\<in>S - it. R y x\<rbrakk> \<Longrightarrow> f x \<sigma> \<le> SPEC (I (it - {x}))"   and "\<And>\<sigma>. \<lbrakk>I {} \<sigma>; c \<sigma>\<rbrakk> \<Longrightarrow> P \<sigma>"   and "\<And>it \<sigma>. \<lbrakk>it \<subseteq> S; I it \<sigma>; \<not> c \<sigma>; \<forall>x\<in>it. \<forall>y\<in>S - it. R y x\<rbrakk> \<Longrightarrow> P \<sigma>" shows "FOREACH\<^sub>O\<^sub>C\<^bsup>R,I\<^esup> S c f \<sigma>0 \<le> SPEC P"
Rep_downsets: fixes x :: "'b downsets" shows "Rep_downsets x \<in> range \<Down>"
subpath_def: fixes a :: "real"   and b :: "real"   and g :: "real \<Rightarrow> 'b" shows "subpath a b g \<equiv> \<lambda>x. g ((b - a) * x + a)"
set_mode_def: shows "set_mode = (\<lambda>shadow_root_ptr. put_M\<^sub>S\<^sub>h\<^sub>a\<^sub>d\<^sub>o\<^sub>w\<^sub>R\<^sub>o\<^sub>o\<^sub>t shadow_root_ptr mode_update)"
ord_isom_surj_to: fixes E :: "('e, 'f) Order_scheme"   and f :: "'a \<Rightarrow> 'e" assumes "Order E"   and "ord_isom D E f" shows "surj_to f (carrier D) (carrier E)"
meth_def_map': fixes list :: "meth_def list" shows "map ((\<lambda>(md, cl, m, vds, mb). md) \<circ> (\<lambda>md. case md of meth_def_def (meth_sig_def cl m vds) mb \<Rightarrow> (md, cl, m, vds, mb))) list = list"
minus_minus: fixes x :: "'c" assumes "x \<in> carrier M" shows "\<ominus>\<^bsub>M\<^esub> (\<ominus>\<^bsub>M\<^esub> x) = x"
fsafe_Not_simps: fixes p :: "('a, 'b, 'c) formula" shows "fsafe (! p) = fsafe p"
Vinv1D: fixes x :: "'b v_state_scheme"   and r :: "nat" assumes "x \<in> Vinv1"   and "next_round x \<le> r" shows "votes x r = Map.empty"
ltype_simps_4: fixes T :: "Arraytype"   and a :: "nat"   and i :: "nat" shows "ltype (arrLoc T a i) = at2jt T"
assms: shows "X #[[y \<and> x]]_zs #= X' #[[y' \<and> x']]_zs'"
m1: shows "mC_C theta c c1 s s1 P1 F1"
not_dvd_imp_multiplicity_0: fixes p :: "'a"   and x :: "'a" assumes "\<not> p dvd x" shows "multiplicity p x = 0"
finite_has_maximal: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "class.order less_eq less"   and "finite A"   and "A \<noteq> {}" shows "\<exists>m\<in>A. \<forall>b\<in>A. less_eq m b \<longrightarrow> m = b"
pochhammer_neq_0_mono: fixes a :: "'a"   and m :: "nat"   and n :: "nat" assumes "pochhammer a m \<noteq> (0::'a)"   and "n \<le> m" shows "pochhammer a n \<noteq> (0::'a)"
a_Case: fixes \<Gamma> :: "(Syntax.var, ty) fmap"   and e :: "Syntax.term"   and eP :: "Syntax.term"   and eV :: "Syntax.term"   and \<tau>\<^sub>1 :: "ty"   and \<tau>\<^sub>2 :: "ty"   and e\<^sub>1 :: "Syntax.term"   and eP\<^sub>1 :: "Syntax.term"   and eV\<^sub>1 :: "Syntax.term"   and \<tau> :: "ty"   and e\<^sub>2 :: "Syntax.term"   and eP\<^sub>2 :: "Syntax.term"   and eV\<^sub>2 :: "Syntax.term" assumes "\<Gamma> \<turnstile> e, eP, eV : Syntax.Sum \<tau>\<^sub>1 \<tau>\<^sub>2"   and "\<Gamma> \<turnstile> e\<^sub>1, eP\<^sub>1, eV\<^sub>1 : Fun \<tau>\<^sub>1 \<tau>"   and "\<Gamma> \<turnstile> e\<^sub>2, eP\<^sub>2, eV\<^sub>2 : Fun \<tau>\<^sub>2 \<tau>" shows "\<Gamma> \<turnstile> Case e e\<^sub>1 e\<^sub>2, Case eP eP\<^sub>1 eP\<^sub>2, Case eV eV\<^sub>1 eV\<^sub>2 : \<tau>"
reachable_mk_symmetric_eq: fixes u :: "'a"   and v :: "'a" assumes "symmetric G" shows "(u \<rightarrow>\<^sup>*\<^bsub>with_proj (mk_symmetric G)\<^esub> v) = (u \<rightarrow>\<^sup>* v)"
sngl_if_invar: fixes l :: "('b \<times> nat) tree"   and a :: "'b"   and n :: "nat"   and r :: "('b \<times> nat) tree" assumes "invar \<langle>l, (a, n), r\<rangle>"   and "n = lvl r" shows "sngl r"
ide_def: fixes a :: "'d" shows "D.ide a \<equiv> a \<cdot>\<^sub>D a \<noteq> D.null \<and> (\<forall>f. (f \<cdot>\<^sub>D a \<noteq> D.null \<longrightarrow> f \<cdot>\<^sub>D a = f) \<and> (a \<cdot>\<^sub>D f \<noteq> D.null \<longrightarrow> a \<cdot>\<^sub>D f = f))"
inverse_arrows_def: fixes f :: "'s"   and g :: "'s" shows "S.inverse_arrows f g \<equiv> S.ide (g \<cdot>\<^sub>S f) \<and> S.ide (f \<cdot>\<^sub>S g)"
bag_finite: fixes t :: "'c" assumes "t \<in> V\<^bsub>T\<^esub>" shows "finite (bag t)"
map_comp: fixes f :: "'d \<Rightarrow> 'c"   and g :: "'e \<Rightarrow> 'd"   and p :: "'e spmf" shows "map_spmf f (map_spmf g p) = map_spmf (f \<circ> g) p"
unit_factor_normalize: fixes a :: "'b" assumes "a \<noteq> (0::'b)" shows "unit_factor (normalize a) = (1::'b)"
U1_swap: shows "U1 ;; swap\<^sub>m = U0"
statesDecls_effect: fixes ef :: "edgeEffect"   and ss :: "(integer \<times> edge list) iarray"   and d :: "procVarDecl" assumes "ef \<in> effect ` edgeSet ss"   and "ef = EEDecl d" shows "d \<in> statesDecls ss"
supp_map_transform_step: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and t :: "'b \<Rightarrow> exp \<Rightarrow> exp"   and ae :: "Vars.var \<Rightarrow> 'b\<^sub>\<bottom>" assumes "\<And>x e a. (x, e) \<in> set \<Gamma> \<Longrightarrow> supp (t a e) \<subseteq> supp e" shows "supp (map_transform t ae \<Gamma>) \<subseteq> supp \<Gamma>"
strict_monoI: 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"   and "\<And>x y. less x y \<Longrightarrow> f x < f y" shows "order.strict_mono less f"
path_components_of_empty_space: fixes X :: "'b topology" assumes "topspace X = {}" shows "path_components_of X = {}"
sep_add_ac_2: fixes x :: "'a"   and y :: "'a" assumes "x ## y" shows "x + y = y + x"
ord0_leI: fixes x :: "ord0"   and y :: "ord0" assumes "\<forall>a<x. a < y" shows "x \<le> y"
interaction_bound_IO: fixes out :: "'e"   and c :: "('d, 'e, 'f) rpv"   and gpv :: "('d, 'e, 'f) gpv"   and "consider" :: "'e \<Rightarrow> bool"   and input :: "'f" assumes "IO out c \<in> set_spmf (the_gpv gpv)" shows "(if consider out then eSuc (interaction_bound consider (c input)) else interaction_bound consider (c input)) \<le> interaction_bound consider gpv"
exclude_one_notin_smap_singleton: fixes s :: "'w"   and w :: "'w" assumes "s \<in> S" shows "w +o \<langle>S - {s}\<rangle> \<notin> dual_order.poset_simplex_map \<P> (w +o \<langle>{s}\<rangle>)"
path_connectedin_def: fixes X :: "'a topology"   and S :: "'a set" shows "path_connectedin X S \<equiv> S \<subseteq> topspace X \<and> path_connected_space (subtopology X S)"
measure_return_pmf: fixes x :: "'b"   and A :: "'b set" shows "measure_pmf.prob (return_pmf x) A = indicat_real A x"
integral_restrict_UNIV: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue" shows "LINT x|lebesgue. (if x \<in> S then f x else (0::'b)) = integral\<^sup>L (lebesgue_on S) f"
matrix_vector_mult_matrix_matrix_mult_compose: fixes A :: "(('b, 'e) vec, 'd) vec"   and B :: "(('b, 'c) vec, 'e) vec" shows "(*v) (A ** B) = (*v) A \<circ> (*v) B"
simps_4: fixes D :: "ast_domain"   and ty_ent :: "'a \<Rightarrow> PDDL_STRIPS_Semantics.type option"   and v :: "'a atom formula" shows "ast_domain.wf_fmla_atom D ty_ent (\<^bold>\<not> v) = False"
rotate_toppos_def: fixes x :: "wall"   and y :: "wall" shows "rotate_toppos x y \<equiv> x = basic [vert, over] \<circ> basic [cap, vert] \<and> y = basic [brick.under, vert] \<circ> basic [vert, cap]"
m: fixes l :: "nat list" shows "(case l of [] \<Rightarrow> 0 | x # l' \<Rightarrow> ack x x) < ack m (sum_list l)"
simps_2: fixes f :: "expr"   and x :: "expr"   and xs :: "expr list" shows "f \<bullet> (x # xs) = f \<cdot> x \<bullet> xs"
iszero_0: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a" assumes "class.ring_1 one times plus zero minus uminus" shows "ring_1.iszero zero zero"
new_tstp_def: shows "new_tstp = (if left I = 0 then Inr tp else Inl (nt - (left I - 1)))"
stmt_step_data_store_sops_range: fixes \<theta> :: "nat \<Rightarrow> nat option"   and s :: "stmt"   and t :: "nat"   and s' :: "stmt"   and t' :: "nat"   and "is" :: "instr list" assumes "\<theta>\<turnstile> (s, t) \<rightarrow>\<^sub>s ((s', t'), is)"   and "valid_sops_stmt t s" shows "\<forall>(D, f)\<in>store_sops is. \<forall>i\<in>D. i < t'"
delivered_to_net_ips: fixes s :: "'c net_state"   and np :: "nat \<Rightarrow> ('c, 'd seq_action) automaton"   and p :: "net_tree"   and i :: "nat"   and d :: "nat"   and s' :: "'c net_state" assumes "s \<in> reachable (pnet np p) TT"   and "(s, i:deliver(d), s') \<in> automaton.trans (pnet np p)" shows "i \<in> net_ips s"
set_disconnected_nodes_get_tag_name: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_disconnected_nodes :: "(_) document_ptr \<Rightarrow> (_) node_ptr list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_tag_name :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, char list) prog"   and get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set"   and ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap"   and ptr' :: "(_) element_ptr" assumes "l_set_disconnected_nodes_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_disconnected_nodes set_disconnected_nodes_locs get_tag_name get_tag_name_locs" shows "\<forall>w\<in>l_set_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_set_disconnected_nodes_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>l_get_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_get_tag_name_locs ptr'. r h h')"
union_eq_empty: fixes M :: "'a multiset"   and N :: "'a multiset" shows "(M + N = {#}) = (M = {#} \<and> N = {#})"
min_absorb2: fixes y :: "'c"   and x :: "'c" assumes "y \<le> x" shows "min x y = y"
prime_factors_lcm: 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 gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and Gcd :: "'a set \<Rightarrow> 'a"   and Lcm :: "'a set \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.factorial_semiring_gcd divide plus minus zero times one unit_factor normalize gcd lcm Gcd Lcm"   and "a \<noteq> zero"   and "b \<noteq> zero" shows "set_mset (factorial_semiring.prime_factorization zero times one normalize (lcm a b)) = set_mset (factorial_semiring.prime_factorization zero times one normalize a) \<union> set_mset (factorial_semiring.prime_factorization zero times one normalize b)"
norm_eq_sqrt_inner: fixes x :: "'b" shows "norm x = sqrt (x \<bullet> x)"
char_mult: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier G"   and "b \<in> carrier G" shows "\<chi> (a \<otimes> b) = \<chi> a * \<chi> b"
oCons: fixes ys :: "'a list" shows "lex_ext_unbounded f as ys = ((\<exists>i<length as. i < length ys \<and> (\<forall>j<i. snd (f (as ! j) (ys ! j))) \<and> fst (f (as ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (as ! i) (ys ! i))) \<and> length ys < length as, (\<exists>i<length as. i < length ys \<and> (\<forall>j<i. snd (f (as ! j) (ys ! j))) \<and> fst (f (as ! i) (ys ! i))) \<or> (\<forall>i<length ys. snd (f (as ! i) (ys ! i))) \<and> length ys \<le> length as)"
leg_simps_3: shows "C.arr \<nu>.dom.leg1"
CollectNestedIteratorTE: fixes \<Gamma> :: "String.literal \<rightharpoonup>\<^sub>f 'a OCL_Types.type"   and src :: "'a expr"   and k :: "call_kind"   and its :: "String.literal list"   and its_ty :: "'a OCL_Types.type option"   and body :: "'a expr"   and \<tau> :: "'a OCL_Types.type" assumes "\<Gamma> \<turnstile>\<^sub>E CollectNestedIteratorCall src k its its_ty body : \<tau>" obtains \<tau>' :: "'a OCL_Types.type" and \<sigma> :: "'a OCL_Types.type" and \<rho> :: "'a OCL_Types.type" and \<upsilon> :: "'a OCL_Types.type" where "k = ArrowCall" and "\<Gamma> \<turnstile>\<^sub>I (src, its, its_ty, body) : (\<tau>', \<sigma>, \<rho>)" and "length its \<le> Suc 0" and "to_nonunique_collection \<tau>' \<upsilon>" and "update_element_type \<upsilon> \<rho> \<tau>"
intros_2: fixes trans :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> 'b"   and sends :: "'a \<Rightarrow> 'b \<Rightarrow> 'c messageValue \<Rightarrow> ('a, 'c) message \<Rightarrow> nat"   and cfg1 :: "('a, 'c, 'b) configuration"   and cfg2 :: "('a, 'c, 'b) configuration"   and msg :: "('a, 'c) message"   and cfg3 :: "('a, 'c, 'b) configuration" assumes "asynchronousSystem.reachable trans sends cfg1 cfg2"   and "asynchronousSystem.steps trans sends cfg2 msg cfg3" shows "asynchronousSystem.reachable trans sends cfg1 cfg3"
DIM_ge_Suc0: shows "Suc 0 \<le> DIM('b)"
insert_DiffM2: fixes x :: "'a"   and M :: "'a multiset" assumes "x \<in># M" shows "M \<ominus> x \<oplus> x = M"
Gramian_determinant_2: fixes k :: "nat" assumes "k \<le> length fs" shows "0 < gs.Gramian_determinant fs k"
intro: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "category_with_binary_products C"   and "category_with_terminal_object C" shows "cartesian_category C"
prob_algebra_def: fixes K :: "'b measure" shows "prob_algebra K = restrict_space (subprob_algebra K) {M. prob_space M}"
IH: shows "finsum_vec TYPE('a) n vs F $ i = (\<Sum>f\<in>F. vs f $ i)"
hyps: fixes r :: "'a poly"   and d :: "'a poly"   and dr :: "nat" shows "local.divide_poly_main (0::'a) 0 r d dr n_ = 0"
p_nz: shows "xs ! p \<noteq> (0::'a)"
intro_spec_refine: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd nres"   and R :: "('d \<times> 'e) set"   and M :: "'e nres" assumes "\<And>x. x \<in> X \<Longrightarrow> f x \<le> \<Down> R M" shows "RES X \<bind> f \<le> \<Down> R M"
split_gallery: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and C :: "'b set"   and D :: "'b set"   and Cs :: "'b set list" assumes "ChamberComplexEndomorphism X f"   and "C \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X"   and "SimplicialComplex.maxsimpchain X (C # Cs @ [D])" shows "\<exists>As A B Bs. A \<in> f \<turnstile> ChamberComplex.chamber_system X \<and> B \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X \<and> C # Cs @ [D] = As @ A # B # Bs"