hcju/isabelleps huggingface dataset:
subsume_def: fixes Ds :: "'a literal list list"   and C :: "'a literal list" shows "subsume Ds C = (\<exists>D\<in>set Ds. subsumes (mset D) (mset C))"
prime_def: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and p :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalization_semidom.prime zero times one normalize p = (comm_semiring_1.prime_elem times one zero p \<and> normalize p = p)"
remove_key_lookup: fixes k0 :: "'b"   and f :: "'b \<Rightarrow>\<^sub>0 'a"   and k :: "'b" shows "lookup (remove_key k0 f) k = (lookup f k when k \<noteq> k0)"
design_points_nempty: shows "\<V> \<times> I \<noteq> {}"
norm_mult: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and inverse :: "'a \<Rightarrow> 'a"   and 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"   and scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and norm :: "'a \<Rightarrow> real"   and sgn :: "'a \<Rightarrow> 'a"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.real_normed_div_algebra divide inverse one times plus zero minus uminus scaleR dist norm sgn uniformity open" shows "norm (times x y) = norm x * norm y"
cons: fixes root :: "(att \<times> char list, att, nat) env"   and x :: "operation"   and root' :: "(att \<times> char list, att, nat) env"   and xs :: "operation list"   and root'' :: "(att \<times> char list, att, nat) env" assumes "root \<midarrow>x\<rightarrow> root'"   and "root' \<Midarrow>xs\<Rightarrow> root''" shows "root \<Midarrow>(x # xs)\<Rightarrow> root''"
flatten_sequence_flatten: fixes c :: "('d, 'e, 'f) com" shows "flatten (sequence Seq (flatten c)) = flatten c"
update_convs_2: fixes doctype' :: "char list \<Rightarrow> char list"   and nothing :: "unit"   and nothinga :: "unit"   and doctype :: "char list"   and document_element :: "(_) element_ptr option"   and disconnected_nodes :: "(_) node_ptr list"   and more :: "'p" shows "doctype_update doctype' \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr> = \<lparr>RObject.nothing = nothing, RDocument.nothing = nothinga, doctype = doctype' doctype, document_element = document_element, disconnected_nodes = disconnected_nodes, \<dots> = more\<rparr>"
minus_equation_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "(uminus a = b) = (uminus b = a)"
advantage_nonneg: fixes \<eta> :: "nat"   and \<A> :: "((bool list \<times> bool list) \<times> 'b, bool list, bool list \<times> bool list) gpv \<times> (bool list \<times> bool list \<Rightarrow> 'b \<Rightarrow> (bool, bool list, bool list \<times> bool list) gpv)" shows "0 \<le> local.ind_cpa.advantage \<eta> \<A>"
wf_minit0: fixes \<phi> :: "'a MFOTL.formula"   and n :: "nat"   and \<sigma> :: "(char list \<times> 'a list) trace"   and R :: "'a list set" assumes "safe_formula \<phi>"   and "\<forall>x\<in>fv \<phi>. x < n" shows "wf_mformula \<sigma> 0 n R (minit0 n \<phi>) \<phi>"
intros_4: fixes x :: "'c"   and G :: "('c, 'd) monoid_scheme"   and gens :: "'c set"   and y :: "'c" assumes "x \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"   and "y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>" shows "x \<otimes>\<^bsub>G\<^esub> y \<in> \<langle>gens\<rangle>\<^bsub>G\<^esub>"
descentE: fixes s :: "'sequent"   and S :: "'structure"   and r :: "'rule"   and s' :: "'sequent"   and S' :: "'structure"   and v :: "'marker"   and b :: "bool"   and v' :: "'marker" assumes "descent (s, S) r (s', S')"   and "(v, b, v') \<in> \<delta> s r s'" shows "(\<sigma> (v', S'), \<sigma> (v, S)) \<in> Ord \<and> (b \<longrightarrow> \<sigma> (v', S') \<noteq> \<sigma> (v, S))"
cinner_minus_right: fixes scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and scaleC :: "complex \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and cinner :: "'b \<Rightarrow> 'b \<Rightarrow> complex"   and x :: "'b"   and y :: "'b" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x (uminus y) = - cinner x y"
path_subset: fixes m :: "nat" assumes "0 < m" shows "path_image (\<gamma> m) \<subseteq> A m - S m"
process_rule_2_failures: fixes xs :: "'a list"   and xs' :: "'a list"   and X :: "'a set"   and P :: "'a process" assumes "(xs @ xs', X) \<in> failures P" shows "(xs, {}) \<in> failures P"
whole_ideal: fixes R :: "('e, 'f) Ring_scheme" assumes "Ring R" shows "ideal R (carrier R)"
zero_less_dist_iff: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.metric_space dist uniformity open" shows "(0 < dist x y) = (x \<noteq> y)"
aligned\<^sub>e_encode: fixes base :: "nat"   and e :: "nat"   and M :: "nat" assumes "2 \<le> base" shows "aligned\<^sub>e base e (encode base e M)"
split_Vertex1_eq_0I: fixes y :: "'v vertex"   and x :: "'v"   and f :: "'v \<times> 'v \<Rightarrow> ennreal" assumes "\<And>z. y \<noteq> Edge x z" shows "split f (Vertex x, y) = 0"
cosafety_fo_spec_axioms: shows "cosafety_fo_spec nfv fv sat"
inf_conj_product: fixes a :: "'a::refinement_lattice"   and b :: "'a::refinement_lattice"   and c :: "'a::refinement_lattice"   and d :: "'a::refinement_lattice" shows "((a::'a::refinement_lattice) \<sqinter> (b::'a::refinement_lattice)) \<iinter> ((c::'a::refinement_lattice) \<sqinter> (d::'a::refinement_lattice)) = a \<iinter> c \<sqinter> a \<iinter> d \<sqinter> b \<iinter> c \<sqinter> b \<iinter> d"
ltl_equivalence_axioms: shows "ltl_equivalence (\<sim>\<^sub>C)"
widen_Class: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and T :: "ty"   and C :: "char list" shows "subtype P T (Class C) = (T = NT \<or> (\<exists>D. T = Class D \<and> P \<turnstile> D \<preceq>\<^sup>* C))"
bl_pad_to_def: fixes bl :: "bool list"   and sz :: "nat" shows "bl_pad_to bl sz \<equiv> bl @ replicate (sz - length bl) False"
srev_srev_ident: fixes xs :: "[:'r:]" shows "srev\<cdot>(srev\<cdot>xs) = xs"
IH: fixes Q :: "(char list \<Rightarrow> int) \<Rightarrow> bool" shows "\<turnstile>\<^sub>t {wp\<^sub>t c Q} c {Q}"
eq_matI: fixes B :: "'c mat"   and A :: "'c mat" assumes "\<And>i j. \<lbrakk>i < dim_row B; j < dim_col B\<rbrakk> \<Longrightarrow> A $$ (i, j) = B $$ (i, j)"   and "dim_row A = dim_row B"   and "dim_col A = dim_col B" shows "A = B"
dom_eq_singleton_conv: fixes f :: "'a \<Rightarrow> 'b option"   and x :: "'a" shows "(dom f = {x}) = (\<exists>v. f = \<epsilon>(x \<mapsto> v))"
sum_list_listset_dirsum: fixes As :: "'a set list"   and as :: "'a list" assumes "add_independentS As"   and "as \<in> listset As" shows "sum_list as \<in> (\<Oplus>A\<leftarrow>As. A)"
order_impl_l: fixes a :: "'a"   and b :: "'a" assumes "a \<le> b" shows "a r\<rightarrow> b = (1::'a)"
runit_commutes_with_R: fixes a :: "'a" assumes "ide a" shows "\<r>\<^sub>1[a \<otimes>\<^sub>1 \<I>\<^sub>1] = \<r>\<^sub>1[a] \<otimes>\<^sub>1 \<I>\<^sub>1"
Range_iff: fixes a :: "'b"   and r :: "('c \<times> 'b) set" shows "(a \<in> Range r) = (\<exists>y. (y, a) \<in> r)"
sum: shows "(\<Sum>x\<in>A. f x *s x) = (0::'b)"
step: fixes stk' :: "'addr val list"   and loc' :: "'addr val list"   and pc' :: "nat"   and xcp' :: "'addr option"   and stk'' :: "'addr val list"   and loc'' :: "'addr val list"   and pc'' :: "nat"   and xcp'' :: "'addr option" assumes "\<tau>Exec_movet ci P t e h (stk, loc, pc, xcp) (stk', loc', pc', xcp')"   and "\<tau>exec_move ci P t e h (stk', loc', pc', xcp') (stk'', loc'', pc'', xcp'')"   and "Q stk' loc' pc' xcp'" shows "Q stk'' loc'' pc'' xcp''"
path_connected_space_iff_components_eq: fixes X :: "'c topology" shows "path_connected_space X = (\<forall>C\<in>path_components_of X. \<forall>C'\<in>path_components_of X. C = C')"
drop_Suc: fixes n :: "nat"   and xs :: "'a list" shows "drop (Suc n) xs = drop n (tl xs)"
nb: fixes f :: "sat_plan_variable formula" assumes "f \<in> set (map (\<lambda>op. Atom (Operator t (index (\<Pi>\<^sub>\<O>) op))) (filter (\<lambda>op. ListMem v (add_effects_of op)) (\<Pi>\<^sub>\<O>)))" shows "is_lit_plus f"
poly_list_set_fml: fixes fml :: "'a fml" shows "set (poly_list fml) = set_fml fml"
r_inv: fixes R :: "('b, 'c) ring_scheme"   and x :: "'b" assumes "field R"   and "x \<in> carrier R"   and "x \<noteq> \<zero>\<^bsub>R\<^esub>" shows "x \<otimes>\<^bsub>R\<^esub> inv\<^bsub>R\<^esub> x = \<one>\<^bsub>R\<^esub>"
primroot_len_mult: fixes u :: "'b list"   and q :: "'b list" assumes "u \<noteq> \<epsilon>"   and "u \<in> q*" obtains k :: "nat" where "\<^bold>|q\<^bold>| = k * \<^bold>|\<rho> u\<^bold>|"
assms: shows "eval the_singleton t t' n (\<box>\<^sub>b\<gamma>)"
hyps_1: shows "e \<turnstile>\<^sub>R s : T \<Rightarrow> U"
PO_s0g_secrecy_trans: shows "{s0g_secrecy} TS.trans s0g {> s0g_secrecy}"
simps: fixes n :: "nat" shows "chan n = (if n = 0 then Some (P, Q) else if n = 1 then Some (Q, P) else None)"
conv_abscissa_newman: shows "conv_abscissa fds_newman \<le> 1"
intros: fixes a :: "'a"   and b :: "'b"   and r :: "('a \<times> 'b) set" assumes "(a, b) \<in> r" shows "b \<in> Range r"
subtuple_fmdom: fixes ym :: "'d \<rightharpoonup>\<^sub>f 'f"   and f :: "'e \<Rightarrow> 'f \<Rightarrow> bool"   and xm :: "'d \<rightharpoonup>\<^sub>f 'e"   and g :: "'f \<Rightarrow> 'e \<Rightarrow> bool" assumes "subtuple f xm ym"   and "subtuple g ym xm" shows "fmdom xm = fmdom ym"
shared_eq: shows "\<S> = \<S>\<^sub>s\<^sub>b"
C\<^sub>i\<^sub>v: fixes y :: "i" shows "cod y \<cdot> y \<cong> y"
simps: fixes a1 :: "real \<Rightarrow> real \<times> real"   and a2 :: "(int \<times> (real \<Rightarrow> real \<times> real)) set" shows "chain_subdiv_path a1 a2 = (\<exists>l \<gamma>. a1 = \<gamma> \<and> a2 = set l \<and> distinct l \<and> rec_join l = \<gamma> \<and> valid_chain_list l)"
atom_map_Inf_pres: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and X :: "'a set" assumes "class.complete_boolean_algebra_alt Inf Sup inf less_eq less sup bot top minus uminus" shows "bounded_lattice.atom_map less_eq less bot (Inf X) = Collect (bounded_lattice.atom less bot) \<inter> \<Inter> (bounded_lattice.atom_map less_eq less bot ` X)"
octo_in_Reals_if_Re: fixes q :: "real" shows "Ree (octo_of_real q) = q"
diff_delta_arctan_upper_31_def: shows "diff_delta_arctan_upper_31 \<equiv> \<lambda>x. 64 / (x\<^sup>2 * (15 + 70 * x\<^sup>2 + 63 * x ^ 4)\<^sup>2 * (1 + x\<^sup>2))"
normalize_1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize" shows "normalize one = one"
in_sccsE: fixes c :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "c \<in> pre_digraph.sccs G" obtains "induced_subgraph c G" and "strongly_connected c" and "\<nexists>d. induced_subgraph d G \<and> strongly_connected d \<and> verts c \<subset> verts d"
assms: shows "map_of \<Gamma> = map_of \<Delta>"
horizontal_chop_leq2: fixes v :: "view"   and u :: "view"   and w :: "view" shows "v=u\<parallel>w \<longrightarrow> w \<le> v"
pred_alt_def_2: fixes cs :: "(Term_Class.term \<times> sterm) list"   and \<Gamma> :: "(name, value) fmap" shows "vclosed (Vabs cs \<Gamma>) = (closed_venv \<Gamma> \<and> list_all (\<lambda>(pat, t). closed_except t (fmdom \<Gamma> |\<union>| frees pat)) cs)"
record_simps_6: fixes f :: "int \<Rightarrow> int"   and x :: "x" shows "a (update_b f x) = a x"
absorb_iff2: fixes a :: "'a"   and b :: "'a" shows "(a \<le> b) = (max a b = b)"
prems_1: shows "rga_ops (pre @ (xs @ [x]) @ [(oid, ref)])"
length_trace_parallel_plan_strips_lte_length_plan_plus_one: fixes I :: "'b \<Rightarrow> bool option"   and \<pi> :: "'b strips_operator list list" shows "length (trace_parallel_plan_strips I \<pi>) \<le> length \<pi> + 1"
measurable_Jordan: fixes S :: "'a set" assumes "bounded S"   and "negligible (frontier S)" shows "S \<in> lmeasurable"
f12: fixes out :: "'c"   and c :: "('a, 'c, 'b) rpv"   and gpv :: "('a, 'c, 'b) gpv"   and input :: "'b"   and x :: "'a" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> (\<I>_uniform UNIV {x. Domainp R x}) out"   and "x \<in> results_gpv (\<I>_uniform UNIV {x. Domainp R x}) (c input)"   and "Domainp (rel_gpv'' A C R) gpv" shows "Domainp (rel_gpv'' A C R) (c input)"
assms_4: shows "\<Z>_sgrp \<alpha> [\<SS>\<lparr>\<A>\<rparr>, \<SS>\<lparr>vmult\<rparr>]\<^sub>\<circ>"
commute: fixes a :: "'d"   and b :: "'d" shows "min a b = min b a"
main: fixes pa :: "'a"   and qa :: "'a"   and p'a :: "'a"   and q'a :: "'a"   and r :: "'a" assumes "coprime pa qa"   and "p'a * pa = q'a * qa"   and "pa \<noteq> (0::'a)"   and "qa \<noteq> (0::'a)"   and "r dvd pa"   and "r dvd qa" shows "pa dvd q'a"
coeff_mpoly_map_vars: fixes f :: "nat \<Rightarrow> nat"   and p :: "'b mpoly"   and mon :: "nat \<Rightarrow>\<^sub>0 nat" assumes "bij f" shows "MPoly_Type.coeff (mpoly_map_vars f p) mon = MPoly_Type.coeff p (permutep f mon)"
v1: shows "viewR t = Some (a, s) \<and> ft_invar s \<and> toList t = toList s @ [a]"
assms_2: shows "I \<subseteq>\<^sub>\<circ> J"
bet__coplanar: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p" assumes "Bet A B C" shows "Coplanar A B C D"
1: shows "E.Ide (Dom (fst fg) \<^bold>\<star> Dom (snd fg))"
MClkReply_def: fixes send :: "state \<Rightarrow> PrIds \<Rightarrow> (memOp, Vals) chan"   and rcv :: "state \<Rightarrow> PrIds \<Rightarrow> (rpcOp, Vals) chan"   and cst :: "state \<Rightarrow> PrIds \<Rightarrow> mClkState"   and p :: "PrIds" shows "MClkReply send rcv cst p = (\<not> $Calling rcv p \<and> $(cst!p) = #clkB \<and> Return send p MClkReplyVal<res<rcv!p>> \<and> (cst!p)$ = #clkA \<and> unchanged (caller rcv!p))"
def_LeastI: fixes a :: "'a"   and P :: "'a \<Rightarrow> bool"   and k :: "'a" assumes "a \<equiv> LEAST x. P x"   and "P k" shows "P a"
IntD2: fixes c :: "'e"   and A :: "'e set"   and B :: "'e set" assumes "c \<in> A \<inter> B" shows "c \<in> B"
x_plus_helpers_4: fixes x :: "nat" shows "x + 7 = Suc (x + 6)"
balls_2: fixes x :: "'a" assumes "x \<in> carrier" shows "open (balls x)"
simps_1: fixes s :: "nat \<Rightarrow> 'a list" shows "inf_last_ti s 0 = s 0"
groupWIs_not_empty_elem: fixes V :: "'b wordinterval list"   and c :: "parts_connection"   and rs :: "'b simple_rule list" assumes "V \<in> set (groupWIs c rs)" shows "V \<noteq> []"
pref_imp_trans: fixes F :: "'b \<Rightarrow> bool"   and G :: "'b \<Rightarrow> bool"   and H :: "'b \<Rightarrow> bool" assumes "|~ F \<longrightarrow> G"   and "|~ G \<longrightarrow> H" shows "|~ F \<longrightarrow> H"
chine_assoc_in_hom: shows "\<guillemotleft>\<nu>\<pi>\<rho>.chine_assoc : HH\<nu>\<pi>\<rho>.chine \<rightarrow>\<^sub>C H\<nu>H\<pi>\<rho>.chine\<guillemotright>"
simps_1: fixes var :: "nat"   and b :: "real mpoly"   and c :: "real mpoly"   and p :: "real mpoly" shows "substInfinitesimalLinear var b c (Eq p) = allZero p var"
disj_com: fixes A :: "'a set"   and p :: "'a"   and q :: "'a" shows "(A \<turnstile> p .\<or>. q) = (A \<turnstile> q .\<or>. p)"
snd_image_Sigma: fixes A :: "'c set"   and B :: "'c \<Rightarrow> 'b set" shows "snd ` Sigma A B = \<Union> (B ` A)"
zmult_neg_neg: fixes i :: "int"   and j :: "int" assumes "i < 0"   and "j < 0" shows "0 < i * j"
fun_eq_iff: fixes f :: "'c \<Rightarrow> 'd"   and g :: "'c \<Rightarrow> 'd" shows "(f = g) = (\<forall>x. f x = g x)"
sup_measure'_def: fixes A :: "'b measure"   and B :: "'b measure" shows "sup_measure' A B = measure_of (space A) (sets A) (\<lambda>X. SUP Y\<in>sets A. emeasure A (X \<inter> Y) + emeasure B (X \<inter> - Y))"
IH: fixes s :: "('St, 'U, 'C) state" assumes "isState s" shows "\<exists>sl. wffp sl \<and> list_all isState sl \<and> hd sl = s \<and> map getGMUserCom (tl sl) = ucl"
simps_1: fixes \<sigma> :: "(char \<Rightarrow> trm option) \<times> (char \<Rightarrow> trm option) \<times> (char \<Rightarrow> fml option) \<times> (char \<Rightarrow> game option)"   and U :: "variable set"   and a :: "char" shows "usubstappp \<sigma> U (Game a) = (case SGames \<sigma> a of None \<Rightarrow> (Denotational_Semantics.allvars, Agame (Game a)) | Some r \<Rightarrow> (U \<union> BVG r, Agame r))"
strict_iff_order: fixes a :: "'t"   and b :: "'t" shows "(a \<prec>\<^sub>t b) = (a \<preceq>\<^sub>t b \<and> a \<noteq> b)"
Sup_upper: fixes x :: "'b"   and A :: "'b set" assumes "x \<in> A" shows "x \<le> Sup A"
induce_reachable_preserves_paths: fixes G :: "('c, 'd) pre_digraph"   and u :: "'c"   and v :: "'c" assumes "wf_digraph G"   and "u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> v" shows "u \<rightarrow>\<^sup>*\<^bsub>G \<restriction> {w. u \<rightarrow>\<^sup>*\<^bsub>G\<^esub> w}\<^esub> v"
notin: shows "\<nexists>k'. cs\<^bsup>path \<sigma>\<^esup> k = cs\<^bsup>path \<sigma>'\<^esup> k'"
map_ident: fixes t :: "'c list" shows "map (\<lambda>x. x) t = t"
nextl_app: fixes M :: "'b nfa"   and Q :: "hf set"   and xs :: "'b list"   and ys :: "'b list" assumes "nfa M" shows "nfa.nextl M Q (xs @ ys) = nfa.nextl M (nfa.nextl M Q xs) ys"
option_rel_Some1: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and x :: "'d"   and y :: "'e option" shows "rel_option A (Some x) y = (\<exists>y'. y = Some y' \<and> A x y')"
assms_2: fixes i :: "nat" assumes "i < nr" shows "f i \<le> nc"
induct: fixes P :: "(fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option) \<Rightarrow> (key \<times> fresh_t) signal \<Rightarrow> bool"   and a0 :: "fid_t \<Rightarrow> (role_t \<times> agent list \<times> atom list) option"   and a1 :: "(key \<times> fresh_t) signal" assumes "\<And>runz A B Kab Nb. P runz (Commit [A, B] (Kab, Nb))"   and "\<And>runz A B Kab Nb. P runz (Running [A, B] (Kab, Nb))"   and "\<And>runz va. P runz (Running [] va)"   and "\<And>runz vb va. P runz (Running [vb] va)"   and "\<And>runz vb vd vf vg va. P runz (Running (vb # vd # vf # vg) va)"   and "\<And>runz va. P runz (Commit [] va)"   and "\<And>runz vb va. P runz (Commit [vb] va)"   and "\<And>runz vb vd vf vg va. P runz (Commit (vb # vd # vf # vg) va)" shows "P a0 a1"
antisym_conv1: fixes x :: "'x \<Rightarrow>\<^sub>0 nat"   and y :: "'x \<Rightarrow>\<^sub>0 nat" assumes "\<not> x \<prec> y" shows "(x \<preceq> y) = (x = y)"
valid_path_trans_observable_moves: fixes m :: "'node"   and as :: "'edge list"   and m' :: "'node"   and cf :: "('var \<Rightarrow> 'val option) \<times> 'ret"   and s' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list"   and S :: "'node SDG_node set" assumes "m -as\<rightarrow>\<^sub>\<surd>* m'"   and "preds (kinds as) [cf]"   and "transfers (kinds as) [cf] = s'" obtains ms'' :: "'node list" and s'' :: "(('var \<Rightarrow> 'val option) \<times> 'ret) list" and ms' :: "'node list" and as' :: "'edge list" and as'' :: "'edge list" where "S,kind \<turnstile> ([m],[cf]) =slice_edges S [] as\<Rightarrow>* (ms'',s'')" and "S,kind \<turnstile> (ms'',s'') =as'\<Rightarrow>\<^sub>\<tau> (m' # ms',s')" and "slice_edges S [] as = slice_edges S [] as''" and "m -as'' @ as'\<rightarrow>\<^sub>\<surd>* m'"
nesting_lemma: fixes P :: "'b set \<Rightarrow> 'b set \<Rightarrow> bool"   and bdy :: "'c \<Rightarrow> 'd set"   and mgf :: "'d \<Rightarrow> 'b"   and f :: "'c \<Rightarrow> 'd"   and U :: "'c set"   and t :: "'d" assumes "\<And>A ts. ts \<subseteq> A \<Longrightarrow> P A ts"   and "\<And>A pn. \<forall>b\<in>bdy pn. P (insert (mgf (f pn)) A) {mgf b} \<Longrightarrow> P A {mgf (f pn)}"   and "\<And>A t. \<forall>pn\<in>U. P A {mgf (f pn)} \<Longrightarrow> P A {mgf t}"   and "finite U" shows "P {} {mgf t}"
k: shows "spr_simAbs k \<in> (\<Union>a. SPRdet.sim_equiv_class a ` SPR.jkbpC)"
gt_wt: fixes s :: "('s, 'v) tm"   and t :: "('s, 'v) tm" assumes "vars_mset s \<subseteq># vars_mset t"   and "local.kbo_std.wt s < local.kbo_std.wt t" shows "local.kbo_std.gt t s"
unfoldr_simps_1: fixes g :: "('a, 'b) generator"   and s :: "'b" assumes "list.has_next g s" shows "list.unfoldr g s = fst (list.next g s) # list.unfoldr g (snd (list.next g s))"
simps_1: fixes \<gamma> :: "('b \<Rightarrow> 'c \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'c \<Rightarrow> bool)"   and a :: "action"   and p :: "'c" shows "match_list \<gamma> [] a p = False"
prems_1: shows "list_all (\<lambda>a. case a of (a, b) \<Rightarrow> (case a of (idx, ty) \<Rightarrow> \<lambda>t. typ_of t = Some ty) b) (x # xs)"
sup_selective: fixes x :: "'c"   and y :: "'c" shows "x \<squnion> y = x \<or> x \<squnion> y = y"
1: shows "(p \<bullet> bn \<alpha>) \<sharp>* Q1"
dim_zero_subspace_eq: fixes A :: "('b, 'c) vec set" assumes "vec.subspace A" shows "(vec.dim A = 0) = (A = {0})"
simps_8: fixes v :: "'b" shows "le_rexp One (Atom v) = True"
add_existing_point: fixes p :: "'a" assumes "p \<in> \<V>" shows "add_point p = \<V>"
is_prefix_Prefixes_subset: fixes q :: "'a list"   and p :: "'a list" assumes "is_prefix q p" shows "Prefixes q \<subseteq> Prefixes p"
enum_length: fixes xs :: "'c list"   and index :: "('c \<Rightarrow> 'd) \<Rightarrow> 'c \<Rightarrow> nat \<Rightarrow> 'd \<Rightarrow> 'd \<Rightarrow> nat"   and key :: "'c \<Rightarrow> 'd"   and n :: "nat"   and mi :: "'d"   and ma :: "'d" shows "length (enum xs index key n mi ma) = n"
intruder_synth_induct: fixes M :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term set"   and t :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term"   and P :: "(('fun, 'atom, 'sets) prot_fun, 'b) Term.term set \<Rightarrow> (('fun, 'atom, 'sets) prot_fun, 'b) Term.term \<Rightarrow> bool" assumes "M \<turnstile>\<^sub>c t"   and "\<And>t. t \<in> M \<Longrightarrow> P M t"   and "\<And>T f. \<lbrakk>length T = arity f; public f; \<And>t. t \<in> set T \<Longrightarrow> M \<turnstile>\<^sub>c t; \<And>t. t \<in> set T \<Longrightarrow> P M t\<rbrakk> \<Longrightarrow> P M (Fun f T)" shows "P M t"
adm_subst: fixes t :: "'d \<Rightarrow> 'e"   and P :: "'e \<Rightarrow> bool" assumes "cont t"   and "adm P" shows "adm (\<lambda>x. P (t x))"
thetaWhileW_WbisT: shows "thetaWhileW \<subseteq> WbisT"
MapsToCompDef: fixes C :: "('g, 'h, 'i) Category_scheme"   and f :: "'h"   and X :: "'g"   and Y :: "'g"   and g :: "'h"   and Z :: "'g" assumes "f maps\<^bsub>C\<^esub> X to Y"   and "g maps\<^bsub>C\<^esub> Y to Z" shows "f \<approx>>\<^bsub>C\<^esub> g"
kpath: shows "is_path (\<pi> \<guillemotleft> k)"
m_closed: fixes x :: "'c"   and y :: "'c" assumes "x \<in> carrier A"   and "y \<in> carrier A" shows "x \<otimes>\<^bsub>A\<^esub> y \<in> carrier A"