hcju/isabelleps huggingface dataset:
strong_induct: fixes P :: "'a \<Rightarrow> lam \<Rightarrow> bool"   and z :: "'a"   and lam :: "lam" assumes "\<And>name z. P z (Var name)"   and "\<And>lam1 lam2 z. \<lbrakk>\<And>z. P z lam1; \<And>z. P z lam2\<rbrakk> \<Longrightarrow> P z (App lam1 lam2)"   and "\<And>name lam z. \<lbrakk>name \<sharp> z; \<And>z. P z lam\<rbrakk> \<Longrightarrow> P z (Lam [name].lam)" shows "P z lam"
simps_64: fixes f1 :: "'e \<Rightarrow> 'd"   and f2 :: "nat set \<Rightarrow> 'e \<Rightarrow> 'd"   and f3 :: "nat \<Rightarrow> 'e \<Rightarrow> 'd"   and f4 :: "nat \<Rightarrow> 'd"   and f5 :: "'e \<Rightarrow> 'd"   and f6 :: "nat \<Rightarrow> 'd"   and f7 :: "'e \<Rightarrow> 'd"   and f8 :: "'d"   and x1 :: "'e" shows "(case broadcast x1 of broadcast x \<Rightarrow> f1 x | groupcast x xa \<Rightarrow> f2 x xa | unicast x xa \<Rightarrow> f3 x xa | \<not>unicast x \<Rightarrow> f4 x | send x \<Rightarrow> f5 x | deliver x \<Rightarrow> f6 x | receive x \<Rightarrow> f7 x | \<tau>\<^sub>s \<Rightarrow> f8) = f1 x1"
simps_10: shows "(extN.Top \<le> extN.Bot) = False"
exhaust_sel: fixes literal :: "'a literal" obtains "literal = Pos (atm_of literal)"   | "literal = Neg (atm_of literal)"
trie_keys_dom_lookup: fixes t :: "'b trie_raw" shows "trie_keys t = dom (trie_lookup t)"
arg_minus_pi2_iff: fixes z :: "complex" shows "(z \<noteq> 0 \<and> arg z = - pi / 2) = (is_imag z \<and> Im z < 0)"
fpower_Inf_pres: fixes f :: "'b \<Rightarrow> 'b"   and i :: "nat" assumes "Inf_pres f" shows "Inf_pres (fpower f i)"
Rep_action_inject: fixes x :: "'d action"   and y :: "'d action" shows "(Rep_action x = Rep_action y) = (x = y)"
integrable_indicator_iff: fixes M :: "'b measure"   and A :: "'b set" shows "integrable M (indicat_real A) = (A \<inter> space M \<in> sets M \<and> emeasure M (A \<inter> space M) < \<infinity>)"
id2: shows "root (2 ^ Suc k) (real d ^ 2 ^ Suc k) = real d"
Bignorabimus_def: fixes \<alpha> :: "game" shows "Bignorabimus \<alpha> = {M. \<forall>I \<omega> X. (\<omega> \<in> game_sem I \<alpha> X) = (\<omega> \<in> game_sem I \<alpha> (selectlike X \<omega> M))}"
ras_cast_to_list_def: fixes s :: "'c array \<times> nat" shows "ras_cast_to_list s \<equiv> let (a, n) = s in rev (take n (list_of_array a))"
span_mono: fixes A :: "'c set"   and B :: "'c set" assumes "A \<subseteq> B" shows "ideal A \<subseteq> ideal B"
WbisT_transC: fixes c :: "('test, 'atom) com"   and d :: "('test, 'atom) com"   and s :: "'state"   and t :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "c \<approx>wT d"   and "s \<approx> t"   and "(c, s) \<rightarrow>c (c', s')" obtains d' :: "('test, 'atom) com" and t' :: "'state" where "(d, t) \<rightarrow>*c (d', t')" and "s' \<approx> t'" and "c' \<approx>wT d'"
asm: shows "(as, x) \<approx>res (=) supp p' (as', x')"
QuoteP_Eats: fixes t1 :: "tm"   and u1 :: "tm"   and t2 :: "tm"   and u2 :: "tm" shows "{QuoteP t1 u1, QuoteP t2 u2} \<turnstile> QuoteP (Eats t1 t2) (Q_Eats u1 u2)"
convex_def: fixes s :: "'a set" shows "convex s = (\<forall>x\<in>s. \<forall>y\<in>s. \<forall>u\<ge>0. \<forall>v\<ge>0. u + v = 1 \<longrightarrow> u *\<^sub>R x + v *\<^sub>R y \<in> s)"
entails_trans: fixes P :: "'a \<Rightarrow> real"   and Q :: "'a \<Rightarrow> real"   and R :: "'a \<Rightarrow> real" assumes "P \<tturnstile> Q"   and "Q \<tturnstile> R" shows "P \<tturnstile> R"
tObsC_absI: fixes t' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and t :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState Trace"   and u :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState"   and v :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" assumes "t' \<in> SPR.jkbpC"   and "tObsC t' = tObsC t"   and "u = tFirst t'"   and "v = tLast t'" shows "(u, v) \<in> tObsC_abs t"
simps_13: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case A of A \<Rightarrow> f1 | B \<Rightarrow> f2 | C \<Rightarrow> f3 | D \<Rightarrow> f4) = f1"
nat_int_comparison_1: fixes a :: "nat"   and b :: "nat" shows "(a = b) = (int a = int b)"
simps_4: fixes x2 :: "('a, 'b) Node" shows "Single x2 \<noteq> Empty"
irreducible\<^sub>d_square_free: fixes p :: "'b poly" assumes "irreducible\<^sub>d p" shows "square_free p"
app_def: fixes i :: "instr"   and P :: "'b prog"   and mxs :: "nat"   and T\<^sub>r :: "ty"   and pc :: "nat"   and mpc :: "nat"   and xt :: "ex_table"   and t :: "(ty list \<times> ty err list) option" shows "app i P mxs T\<^sub>r pc mpc xt t = (case t of None \<Rightarrow> True | \<lfloor>\<tau>\<rfloor> \<Rightarrow> app\<^sub>i (i, P, pc, mxs, T\<^sub>r, \<tau>) \<and> xcpt_app i P pc mxs xt \<tau> \<and> (\<forall>(pc', \<tau>')\<in>set (eff i P pc xt t). pc' < mpc))"
Ri_effective: fixes \<Gamma> :: "'b inference set"   and \<gamma> :: "'b inference"   and N :: "'b literal multiset set" assumes "standard_redundancy_criterion_reductive \<Gamma>"   and "\<gamma> \<in> \<Gamma>"   and "concl_of \<gamma> \<in> N \<union> standard_redundancy_criterion.Rf N" shows "\<gamma> \<in> standard_redundancy_criterion.Ri \<Gamma> N"
0: shows "prv (eql (pls (Num m) (suc (Num n))) (suc (pls (Num m) (Num n))))"
free_gen_mem: fixes P :: "('e \<times> 'f, 'a, 'g) Module_scheme"   and M :: "('e, 'a, 'g) Module_scheme"   and N :: "('f, 'a, 'g) Module_scheme"   and a :: "'e \<times> 'f" assumes "FM\<^bsub>R\<^esub> P M N"   and "a \<in> M \<times>\<^sub>c N" shows "a \<in> carrier P"
rqrm_invar_def: fixes rqrm :: "('a, 'b dlist) HashMap.hashmap" shows "rqrm_invar rqrm \<equiv> hm.invar rqrm \<and> (\<forall>q. ls.invar (rqrm_lookup rqrm q))"
vb_not_zero: shows "v $ b \<noteq> (0::'a)"
int_nat_div: fixes a :: "nat"   and b :: "nat" shows "int a div int b = int (a div b)"
"1_2": shows "(\<^bold>\<not> \<^bold>\<forall> p) \<in> S"
Block1Red: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e :: "'addr expr1"   and h :: "'heap"   and x :: "'addr val list"   and ta :: "('addr, 'thread_id, 'heap) external_thread_action"   and e' :: "'addr expr1"   and h' :: "'heap"   and x' :: "'addr val list"   and V :: "nat"   and T :: "ty" assumes "uf,P,t \<turnstile>1 \<langle>e,(h, x)\<rangle> -ta\<rightarrow> \<langle>e',(h', x')\<rangle>" shows "uf,P,t \<turnstile>1 \<langle>{V:T=None; e},(h, x)\<rangle> -ta\<rightarrow> \<langle>{V:T=None; e'},(h', x')\<rangle>"
group_card_in_sizes: fixes g :: "'a set" assumes "g \<in> \<G>" shows "card g \<in> group_sizes"
perp_in_right_comm: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and X :: "'a"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B C D" shows "Tarski_neutral_dimensionless.PerpAt Bet Cong X A B D C"
plus_inverse_ge_2: fixes x :: "real" assumes "0 < x" shows "2 \<le> x + inverse x"
E_validD_1: fixes v :: "'a"   and e :: "'b"   and v' :: "'a" assumes "(v, e, v') \<in> G.E" shows "v \<in> G.V"
errMOD_igAbsIPresIGWlsSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igAbsIPresIGWls MOD"   and "igWlsAbsDisj MOD"   and "igWlsAbsIsInBar MOD" shows "igAbsIPresIGWlsSTR (errMOD MOD)"
disc_37: fixes x2 :: "('d, 'e) Term.term" shows "\<not> is_InSet (receive\<langle>x2\<rangle>)"
proots_count_of_root: fixes P :: "'a poly"   and x :: "'a"   and S :: "'a set" assumes "P \<noteq> 0"   and "x \<in> S"   and "poly P x = (0::'a)" shows "0 < proots_count P S"
timpl_closure'_step_inv: fixes t :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and s :: "(('a, 'b, 'c) prot_fun, 'd) Term.term"   and TI :: "('c set \<times> 'c set) set" assumes "(t, s) \<in> timpl_closure'_step TI" obtains a :: "'c set" and b :: "'c set" where "(a, b) \<in> TI" and "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) t s"
prems_2: shows "P,E \<turnstile> \<langle>es,s\<rangle> [\<Rightarrow>] \<langle>map Val (v # vs),(h, l)\<rangle>"
compile_mod2: fixes n :: "nat"   and ap :: "abc_inst list" shows "is_even (length (concat (take n (tms_of ap))))"
matches_def: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and m :: "'a match_expr"   and a :: "action"   and p :: "'b" shows "Matching_Ternary.matches \<gamma> m a p \<equiv> ternary_to_bool_unknown_match_tac (snd \<gamma>) a p (ternary_ternary_eval (map_match_tac (fst \<gamma>) p m))"
Span_closed: fixes ms :: "'v list" assumes "set ms \<subseteq> V" shows "RSpan ms \<subseteq> V"
AtCondExpr2: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "'d \<Rightarrow> ('e, 'd, 'f) ann list option"   and a2 :: "('e, 'd, 'f) ann"   and c2 :: "('e, 'd, 'f) com"   and a :: "'e set \<times> ('e, 'd, 'f) com"   and r :: "'e set"   and a1 :: "('e, 'd, 'f) ann"   and b :: "'e set"   and c1 :: "('e, 'd, 'f) com" assumes "atomicsR \<Gamma> \<Theta> a2 c2 a" shows "atomicsR \<Gamma> \<Theta> (AnnBin r a1 a2) (Cond b c1 c2) a"
App_eq_foldl_conv: fixes r :: "lam"   and s :: "lam"   and t :: "lam"   and ts :: "lam list" shows "(r \<degree> s = t \<degree>\<degree> ts) = (if ts = [] then r \<degree> s = t else \<exists>ss. ts = ss @ [s] \<and> r = t \<degree>\<degree> ss)"
simps_2: fixes x21 :: "'b"   and x22 :: "'b tree"   and x23 :: "'b tree"   and x24 :: "nat" shows "ET \<noteq> MKT x21 x22 x23 x24"
D: shows "D = v\<^sup>2 - 2 * a * s0"
nat_diff_add_eq1: fixes j :: "nat"   and i :: "nat"   and u :: "nat"   and m :: "nat"   and n :: "nat" assumes "j \<le> i" shows "i * u + m - (j * u + n) = (i - j) * u + m - n"
simps_28: fixes vb :: "bexp"   and vc :: "'b acom"   and vd :: "'b acom"   and ve :: "'b"   and v :: "'b acom"   and va :: "'b acom" shows "(IF vb THEN vc ELSE vd {ve} \<sqsubseteq> v;; va) = False"
disc_1: fixes x11 :: "'a"   and x12 :: "'b"   and x13 :: "'b" shows "isTrans (Trans x11 x12 x13)"
hom_add_eq_zero: fixes x :: "'a"   and y :: "'a" assumes "x + y = (0::'a)" shows "hom x + hom y = (0::'b)"
inv_state: fixes ot :: "'op rose_tree" assumes "local.wf ot" shows "inv (state ot)"
distinct_23: shows "cospan_shape.BB \<noteq> cospan_shape.BT"
irr: fixes f :: "int poly" assumes "f \<in># fs" shows "q.irreducible\<^sub>d_m f"
double_add_less_zero_iff_single_less_zero: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "less (plus a a) zero = less a zero"
assms: fixes fa :: "floatarith" assumes "fa \<in> set fas" shows "freshs_floatarith fa xs"
inv_solve_right': fixes a :: "nat \<Rightarrow> int"   and b :: "nat \<Rightarrow> int"   and c :: "nat \<Rightarrow> int" assumes "a \<in> carrier Zp"   and "b \<in> carrier Zp"   and "c \<in> carrier Zp" shows "(b \<oplus> \<ominus> c = a) = (b = a \<oplus> c)"
fr_out_3: shows "frontier (outside (path_image c1 \<union> path_image c)) = path_image c1 \<union> path_image c"
put_inef: fixes x :: "'d \<Longrightarrow> 'e"   and \<sigma> :: "'e"   and v :: "'d" assumes "ief_lens x" shows "put\<^bsub>x\<^esub> \<sigma> v = \<sigma>"
cupcake_pmatch_preserve: fixes static_cenv :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and pat :: "pat"   and v0 :: "v"   and env :: "(char list \<times> v) list"   and envC :: "(char list, char list, nat \<times> tid_or_exn) namespace"   and env' :: "(char list \<times> v) list" assumes "cakeml_static_env static_cenv"   and "is_cupcake_pat pat"   and "cakeml_static_env.is_cupcake_value static_cenv v0"   and "list_all (cakeml_static_env.is_cupcake_value static_cenv \<circ> snd) env"   and "cupcake_c_ns envC"   and "cupcake_pmatch envC pat v0 env = Match env'" shows "list_all (cakeml_static_env.is_cupcake_value static_cenv \<circ> snd) env'"
sum_mult_ring_multiplication: fixes A :: "'a set"   and B :: "'a set"   and r :: "'a"   and a :: "'a" assumes "A \<subseteq> carrier R"   and "ideal R B"   and "r \<in> carrier R"   and "a \<in> sum_mult R A B" shows "r \<cdot>\<^sub>r a \<in> sum_mult R A B"
nfoldli_simps_2: fixes x :: "'d"   and ls :: "'d list"   and c :: "'c \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c nres"   and s :: "'c" shows "nfoldli (x # ls) c f s = (if c s then f x s \<bind> nfoldli ls c f else RETURN s)"
4: shows "trg \<lparr>Chn = fgh.chine_assoc, Dom = Dom ((f \<star> g) \<star> h), Cod = Cod (f \<star> g \<star> h)\<rparr> = trg f"
set_map_spmf: fixes f :: "'d \<Rightarrow> 'c"   and p :: "'d spmf" shows "set_spmf (map_spmf f p) = f ` set_spmf p"
cycle_free_up_to_diag: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and k :: "nat"   and n :: "nat"   and i :: "nat" assumes "cycle_free_up_to m k n"   and "i \<le> n" shows "\<one> \<le> m i i"
conj_left_commute: fixes b :: "'c"   and a :: "'c"   and c :: "'c" shows "b AND a AND c = a AND b AND c"
r''_1: shows "r'' = Resumption.resumption.resume \<r>'' inp"
sort_ex_def: fixes rel :: "('a \<times> 'a) set"   and S :: "'a set" shows "sort_ex rel S = (S \<subseteq> Field rel)"
inject: fixes x1 :: "int"   and y1 :: "int" shows "(Const x1 = Const y1) = (x1 = y1)"
face_of_halfspace_ge: fixes F :: "'b set"   and b :: "real"   and a :: "'b" shows "(F face_of {x. b \<le> a \<bullet> x}) = (F = {} \<or> F = {x. a \<bullet> x = b} \<or> F = {x. b \<le> a \<bullet> x})"
TarskiAbsolute_axioms: shows "TarskiAbsolute p_congruent p_between"
anno_map_acom: fixes p :: "nat"   and C :: "'a acom"   and f :: "'a \<Rightarrow> 'b" assumes "p < length (annos C)" shows "anno (ACom.map_acom f C) p = f (anno C p)"
identity_arrow_of_spans_axioms: shows "identity_arrow_of_spans (\<cdot>) \<pi>"
prems_3: shows "set (y # ys) \<subseteq> set (xs @ [x])"
weak_bounded_lattice_def: fixes L :: "('c, 'd) gorder_scheme" shows "weak_bounded_lattice L \<equiv> weak_lattice L \<and> weak_partial_order_bottom L \<and> weak_partial_order_top L"
foundation16'': fixes \<tau> :: "'a state \<times> 'a state"   and X :: "'a state \<times> 'a state \<Rightarrow> 'b" shows "(\<tau> \<Turnstile> \<delta> X) \<noteq> (\<tau> \<Turnstile> X \<triangleq> invalid \<or> \<tau> \<Turnstile> X \<triangleq> null)"
ag_neq_diffnonzero: fixes A :: "('c, 'd) aGroup_scheme"   and a :: "'c"   and b :: "'c" assumes "aGroup A"   and "a \<in> carrier A"   and "b \<in> carrier A" shows "(a \<noteq> b) = (a \<plusminus>\<^bsub>A\<^esub> -\<^sub>a\<^bsub>A\<^esub> b \<noteq> \<zero>\<^bsub>A\<^esub>)"
uleu: fixes i :: "nat"   and j :: "nat" assumes "i \<le> j" shows "u i \<le> u j"
zero_into_ordering: fixes f :: "nat \<Rightarrow> 'b"   and betw :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> bool"   and X :: "'b set" assumes "TernaryOrdering.ordering f betw X"   and "X \<noteq> {}" shows "f 0 \<in> X"
connected_eq_connected_components_eq: fixes s :: "'b set" shows "connected s = (\<forall>c\<in>components s. \<forall>c'\<in>components s. c = c')"
simps_1: fixes s :: "'s"   and t_end :: "'s" shows "viterbi s t_end [] = ([], indicator {t_end} s)"
IH: fixes p :: "nat \<Rightarrow> 'a" assumes "0 < deg R p"   and "p \<in> carrier P"   and "Q (Cring_Poly.truncate R p)" shows "Q p"
suc_not_zero: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and one :: "'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and Rep :: "'b \<Rightarrow> int"   and Abs :: "int \<Rightarrow> 'b"   and a :: "'b" assumes "class.mod_type times minus plus zero uminus one less_eq less Rep Abs"   and "mod_type.to_nat Rep a + 1 \<noteq> CARD('b)" shows "plus a one \<noteq> zero"
l6_16_1: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and P :: "'a"   and Q :: "'a"   and S :: "'a"   and X :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "P \<noteq> Q"   and "Tarski_neutral_dimensionless.Col Bet S P Q"   and "Tarski_neutral_dimensionless.Col Bet X P Q" shows "Tarski_neutral_dimensionless.Col Bet X P S"
arc_univalent: fixes x :: "'c \<times> 'c \<Rightarrow> 'd" assumes "matrix_stone_relation_algebra.arc x" shows "matrix_stone_relation_algebra.univalent x"
Union_connected_components_of: fixes X :: "'b topology" shows "\<Union> (connected_components_of X) = topspace X"
poly_list_rel_empty_iff: fixes r :: "(char list multiset \<times> int) multiset"   and R :: "('a \<times> char list multiset) set" shows "(([], r) \<in> poly_list_rel R) = (r = {#})"
red_n_Rec: fixes c :: "com"   and s :: "char list \<Rightarrow> val option"   and c'' :: "com"   and s'' :: "char list \<Rightarrow> val option"   and n :: "nat"   and c' :: "com"   and s' :: "char list \<Rightarrow> val option" assumes "\<langle>c,s\<rangle> \<rightarrow> \<langle>c'',s''\<rangle>"   and "\<langle>c'',s''\<rangle> \<rightarrow>\<^bsup>n\<^esup> \<langle>c',s'\<rangle>" shows "\<langle>c,s\<rangle> \<rightarrow>\<^bsup>Suc n\<^esup> \<langle>c',s'\<rangle>"
freeword_funlift_Abs_freeword: fixes xs :: "('a \<times> bool) list"   and f :: "'a \<Rightarrow> 'b" assumes "proper_signed_list xs" shows "freeword_funlift f (Abs_freeword xs) = sum_list (map (apply_sign f) xs)"
2: shows "(\<Union>f\<in>injections X Y. {f \<union> {(x, y)} |y. y \<in> Y - Range f}) = \<Union> {{f \<union> {(x, y)} |y. y \<in> Y - Range f} |f. f \<in> injections X Y \<and> x \<notin> Domain f}"
same_fw_behaviour_one_def: fixes ip1 :: "'c word"   and ip2 :: "'c word"   and c :: "'d parts_connection_scheme"   and rs :: "'c simple_rule list" shows "same_fw_behaviour_one ip1 ip2 c rs \<equiv> \<forall>d s. runFw ip1 d c rs = runFw ip2 d c rs \<and> runFw s ip1 c rs = runFw s ip2 c rs"
simps: fixes p :: "'a Point"   and cone :: "'a Cone" shows "onCone p cone = (space2 (vertex cone) p = slope cone * slope cone * time2 (vertex cone) p)"
AAcc_\<tau>execI2: fixes ci :: "('addr, 'heap) check_instr"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and e' :: "'addr expr1"   and h :: "'heap"   and 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"   and e :: "'addr expr1"   and v :: "'addr val" assumes "\<tau>exec_move ci P t e' h (stk, loc, pc, xcp) (stk', loc', pc', xcp')" shows "\<tau>exec_move ci P t (e\<lfloor>e'\<rceil>) h (stk @ [v], loc, length (compE2 e) + pc, xcp) (stk' @ [v], loc', length (compE2 e) + pc', xcp')"
par_id: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "A B Par A C" shows "Col A B C"
assms_2: shows "\<bar>a\<bar> \<le> D"
orthogonal_def: fixes x :: "'a"   and y :: "'a" shows "RV_inner.orthogonal x y = (x \<bullet> y = 0)"
expunge_store_commute: fixes i1 :: "'c"   and mo2 :: "'c"   and i2 :: "'c"   and mo1 :: "'c"   and e1 :: "'d"   and e2 :: "'d" assumes "i1 \<noteq> mo2"   and "i2 \<noteq> mo1" shows "\<langle>Expunge e1 mo1 i1\<rangle> \<rhd> \<langle>Store e2 mo2 i2\<rangle> = \<langle>Store e2 mo2 i2\<rangle> \<rhd> \<langle>Expunge e1 mo1 i1\<rangle>"
lm025: fixes P1 :: "'b set set"   and P2 :: "'b set set"   and X :: "'b set"   and Y :: "'b set" assumes "\<Union> P1 \<inter> \<Union> P2 = {}"   and "is_non_overlapping P1"   and "is_non_overlapping P2"   and "X \<in> P1 \<union> P2"   and "Y \<in> P1 \<union> P2"   and "X \<inter> Y \<noteq> {}" shows "X = Y"
merge_tailrec_inefficient_prepend: fixes as :: "'a list"   and bs :: "'a list"   and a :: "'a"   and accs :: "'a list" shows "merge_tailrec_inefficient as bs (a # accs) = a # merge_tailrec_inefficient as bs accs"
divide_inverse_commute: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and a :: "'b"   and b :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide a b = times (inverse b) a"
carrier_mult_of: fixes R :: "('c, 'd) ring_scheme" shows "carrier (Multiplicative_Group.mult_of R) = carrier R - {\<zero>\<^bsub>R\<^esub>}"
alphaAll_preserves_qGoodAll1: fixes X :: "('f, 'g, 'e, 'd, 'h) qTerm"   and A :: "('f, 'g, 'e, 'd, 'h) qAbs" assumes "FixVars TYPE('d) TYPE('e)" shows "(FixVars.qGood X \<longrightarrow> (\<forall>X'. X #= X' \<longrightarrow> FixVars.qGood X')) \<and> (FixVars.qGoodAbs A \<longrightarrow> (\<forall>A'. A $= A' \<longrightarrow> FixVars.qGoodAbs A'))"
s: shows "r = s + x *\<^sub>R (t - s)"
distinct_path_intro: fixes G :: "('a, 'b) graph"   and x1 :: "'a"   and x2 :: "'b"   and x3 :: "'a"   and v :: "'a"   and ps :: "('a \<times> 'b \<times> 'a) list"   and v' :: "'a" assumes "valid_unMultigraph G"   and "valid_unMultigraph.is_trail (del_unEdge x1 x2 x3 G) v ps v'" shows "valid_unMultigraph.is_trail G v ps v'"
le_supI1: fixes sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.semilattice_sup sup less_eq less"   and "less_eq x a" shows "less_eq x (sup a b)"
Server_sent_WL5: fixes B :: "agent"   and A :: "agent"   and NB :: "msg"   and evs :: "event list" assumes "Says Server B (Crypt (shrK B) \<lbrace>Agent A, NB\<rbrace>) \<in> set evs"   and "evs \<in> woolam" shows "\<exists>B'. Says B' Server \<lbrace>Agent A, Agent B, Crypt (shrK A) NB\<rbrace> \<in> set evs"
natural_isomorphism_axioms: shows "natural_isomorphism (\<cdot>) (\<cdot>) (\<lambda>f. local.Prod (cod \<iota>, f)) local.map \<l>.map"
sublens'_def: fixes X :: "'e \<Longrightarrow> 'f"   and Y :: "'g \<Longrightarrow> 'f" shows "X \<subseteq>\<^sub>L' Y = (\<forall>s\<^sub>1 s\<^sub>2 s\<^sub>3. s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>3 = s\<^sub>1 \<triangleleft>\<^bsub>Y\<^esub> (s\<^sub>2 \<triangleleft>\<^bsub>X\<^esub> s\<^sub>3))"
finite_path_has_ends: fixes Q :: "'a set"   and X :: "'a set" assumes "Q \<in> \<P>"   and "X \<subseteq> Q"   and "finite X"   and "3 \<le> card X" shows "\<exists>a\<in>X. \<exists>b\<in>X. a \<noteq> b \<and> (\<forall>c\<in>X. a \<noteq> c \<and> b \<noteq> c \<longrightarrow> [[a c b]])"
p_gt_0: fixes p :: "real"   and M :: "bool stream measure" assumes "infinite_coin_toss_space p M" shows "0 \<le> p"
scalar_invertible_iff: fixes k :: "real"   and A :: "(('a, 'b) vec, 'c) vec" assumes "k \<noteq> 0"   and "invertible A" shows "invertible (k *\<^sub>R A) = (k \<noteq> 0 \<and> invertible A)"
assms_2: shows "\<forall>x. (x, x) \<notin> r"
SUP_cong_simp: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c \<Rightarrow> 'a"   and D :: "'c \<Rightarrow> 'a" assumes "A = B"   and "\<And>x. x \<in> B =simp=> C x = D x" shows "\<Squnion> (C ` A) = \<Squnion> (D ` B)"
ext_hd_or_tl_axioms: shows "ext_hd_or_tl len_lexext_rev"
hyps_1: shows "P,E \<turnstile> \<langle>\<lparr>C\<rparr>e,s\<rangle> \<rightarrow>* \<langle>e'',s''\<rangle>"
wpt_comp: fixes one :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dual :: "'c \<Rightarrow> 'c"   and dual_star :: "'c \<Rightarrow> 'c"   and omega :: "'c \<Rightarrow> 'c"   and star :: "'c \<Rightarrow> 'c"   and bot :: "'c"   and top :: "'c"   and x :: "'c" assumes "class.mbt_algebra one times inf less_eq less sup dual dual_star omega star bot top" shows "times (mbt_algebra.wpt one times inf top x) x = x"
triple_valid_def: fixes n :: "nat"   and t :: "'b triple" shows "|=n:t = (case t of {P}. c .{Q} \<Rightarrow> \<forall>Z s. P Z s \<longrightarrow> (\<forall>s'. <c,s> -n-> s' \<longrightarrow> Q Z s'))"