hcju/isabelleps huggingface dataset:
fresh_fun_eqvt_app: fixes f :: "'a \<Rightarrow> 'b"   and a :: "atom"   and x :: "'a" assumes "eqvt f"   and "a \<sharp> x" shows "a \<sharp> f x"
simps_2: fixes ipassmt :: "iface \<Rightarrow> ('a word \<times> nat) list option"   and ifce :: "iface" shows "iiface_rewrite ipassmt (Match (IIface ifce)) = ipassmt_iface_replace_srcip_mexpr ipassmt ifce"
o_assoc: fixes f :: "'g \<Rightarrow> 'f"   and g :: "'h \<Rightarrow> 'g"   and h :: "'e \<Rightarrow> 'h" shows "f \<circ> (g \<circ> h) = f \<circ> g \<circ> h"
par_strict_not_col_1: 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 C :: "'a"   and D :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.ParStrict Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Col Bet A B C"
VPow_bottom: fixes B :: "V" shows "0 \<in>\<^sub>\<circ> VPow B"
weakly_reduced_def: fixes \<alpha> :: "rat"   and k :: "nat" shows "fs.gs.weakly_reduced \<alpha> k = (\<forall>i. Suc i < k \<longrightarrow> \<parallel>fs.gs.gso i\<parallel>\<^sup>2 \<le> \<alpha> * \<parallel>fs.gs.gso (Suc i)\<parallel>\<^sup>2)"
scale_def: fixes a :: "'a"   and v :: "'b \<times> 'c" shows "scale a v = (s1 a (fst v), s2 a (snd v))"
add_iso: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<le> y" shows "z + x \<le> z + y"
image_insert: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and B :: "'b set" shows "f ` (a \<triangleright> B) = f a \<triangleright> f ` B"
lcf_pow: fixes p :: "nat \<Rightarrow> 'a"   and n :: "nat" assumes "p \<in> carrier P" shows "(p [^]\<^bsub>P\<^esub> n) (deg R (p [^]\<^bsub>P\<^esub> n)) = p (deg R p) [^] n"
fofu_def: shows "fofu \<equiv> let f\<^sub>0 = \<lambda>_. 0::'capacity in while\<^bsup>fofu_invar\<^esup> (\<lambda>(f, brk). \<not> brk) (\<lambda>(f, uu_). find_augmenting_spec f \<bind> (\<lambda>p. case p of None \<Rightarrow> return (f, True) | Some p \<Rightarrow> assert (p \<noteq> []) \<bind> (\<lambda>_. assert (NPreflow.isAugmentingPath c s t f p) \<bind> (\<lambda>_. let f = NFlow.augment_with_path c f p in assert (NFlow c s t f) \<bind> (\<lambda>_. return (f, False)))))) (f\<^sub>0, False) \<bind> (\<lambda>(f, uu_). assert (NFlow c s t f) \<bind> (\<lambda>_. return f))"
list_all2_same: fixes P :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and xs :: "'e list" shows "list_all2 P xs xs = (\<forall>x\<in>set xs. P x x)"
rem_deg: fixes f :: "nat \<Rightarrow> 'a"   and a :: "'a" assumes "f \<in> carrier P"   and "a \<in> carrier R" shows "deg R (UP_root_rem f a) = 0"
assms_2: shows "theta\<inverse> \<subseteq> ZOretrT (theta\<inverse> \<union> ZObisT)"
vars\<^sub>s\<^sub>t\<^sub>pI: fixes x :: "'c"   and s :: "('d, 'c) strand_step" assumes "x \<in> fv\<^sub>s\<^sub>t\<^sub>p s" shows "x \<in> vars\<^sub>s\<^sub>t\<^sub>p s"
sorted_sublist_wrt_cons': fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and xs :: "'c list"   and lo :: "nat"   and hi :: "nat" assumes "sorted_sublist_wrt R xs (lo + 1) hi"   and "lo \<le> hi"   and "hi < length xs"   and "\<forall>j. lo < j \<and> j \<le> hi \<longrightarrow> R (xs ! lo) (xs ! j)" shows "sorted_sublist_wrt R xs lo hi"
zero_ikA: shows "is_zero_row_upt_k i k A"
PO_rhoare_def: fixes pre :: "('e \<times> 'f) set"   and Ra :: "('e \<times> 'e) set"   and Rc :: "('f \<times> 'f) set"   and post :: "('e \<times> 'f) set" shows "{pre} Ra, Rc {> post} \<equiv> pre O Rc \<subseteq> Ra O post"
weakPsiCongParNil: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi" shows "\<Psi> \<rhd> P \<parallel> \<zero> \<doteq> P"
assms_3: fixes a :: "'b \<times> 'c"   and b :: "'b \<times> 'c" shows "(fst (f a) = fst (f b)) = (fst a = fst b)"
SUP1_E: fixes B :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'c set"   and b :: "'b" assumes "Sup (B ` A) b" obtains x :: "'c" where "x \<in> A" and "B x b"
map_add_comm: fixes m1 :: "'a \<Rightarrow> 'b option"   and m2 :: "'a \<Rightarrow> 'b option" assumes "dom m1 \<inter> dom m2 = {}" shows "m2 \<Oplus> m1 = m1 \<Oplus> m2"
ts_inv_ok_conv_dom_eq: fixes ts :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow>f nat) option"   and I :: "'f \<Rightarrow> 'i option" shows "ts_inv_ok ts I = (dom ts = dom I)"
obsf_attach: fixes \<I> :: "('y, 'z) \<I>"   and \<I>' :: "('aa, 'ab) \<I>"   and conv :: "('y, 'z, 'aa, 'ab) converter"   and res :: "('aa, 'ab) resource" assumes "pfinite_converter \<I> \<I>' conv"   and "\<I>, \<I>' \<turnstile>\<^sub>C conv \<surd>"   and "\<I>' \<turnstile>res res \<surd>" shows "outs_\<I> \<I> \<turnstile>\<^sub>R obsf_converter conv \<rhd> obsf_resource res \<sim> obsf_resource (conv \<rhd> res)"
simps_2: fixes x :: "V"   and xs :: "V list" shows "vfsequence_of_vlist (x # xs) = vfsequence_of_vlist xs #\<^sub>\<circ> x"
preserve_coincide1: assumes "vid1 \<notin> ODE_vars I (OVar vid1)" shows "sterm_sem I (f1 fid2 vid1) (fst (mk_v I (OVar vid1) (sol 0, b) (sol t))) = sterm_sem I (f1 fid2 vid1) (sol 0)"
injectivity_image: fixes f :: "'b \<Rightarrow> 'a"   and A :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and invert :: "'a \<Rightarrow> 'b" assumes "f ` A = g ` A"   and "\<forall>x\<in>A. invert (f x) = x \<and> invert (g x) = x" shows "\<forall>x\<in>A. f x = g x"
simps_5: fixes c :: "int"   and e :: "MIR.num" shows "\<zeta> (Lt (CN 0 c e)) = c"
connect_eq_resource_cong: fixes \<I> :: "('a, 'b) \<I>"   and distinguisher :: "('a, 'b) distinguisher"   and res :: "('a, 'b) resource"   and res' :: "('a, 'b) resource" assumes "\<I> \<turnstile>g distinguisher \<surd>"   and "outs_\<I> \<I> \<turnstile>\<^sub>R res \<sim> res'"   and "\<I> \<turnstile>res res \<surd>" shows "connect distinguisher res = connect distinguisher res'"
separating_cut: fixes S :: "('v \<times> 'v) set" assumes "separating \<Gamma> S"   and "source \<Delta> \<noteq> sink \<Delta>"   and "\<And>x. \<not> edge \<Delta> (sink \<Delta>) x" shows "cut \<Delta> (local.cut_of_sep S)"
LLL_invariant_def: fixes upw :: "bool"   and i :: "nat"   and fs :: "int vec list" shows "LLL_invariant upw i fs = (gs.lin_indpt_list (map of_int_hom.vec_hom fs) \<and> lattice_of fs = L \<and> gram_schmidt_fs.reduced n (map of_int_hom.vec_hom fs) \<alpha> i \<and> i \<le> m \<and> length fs = m \<and> (upw \<or> \<mu>_small fs i))"
min_exists: fixes xs :: "('c, 'd) bintree option list"   and a :: "'c" assumes "Binomial_Queue.min xs = Some a" shows "Some a \<in> map_option priority ` set xs"
weak_refl: fixes L :: "('a, 'b) gorder_scheme"   and x :: "'a"   and y :: "'a" assumes "weak_partial_order L"   and "x .=\<^bsub>L\<^esub> y"   and "x \<in> carrier L"   and "y \<in> carrier L" shows "x \<sqsubseteq>\<^bsub>L\<^esub> y"
bounded_linear_def: fixes f :: "'a \<Rightarrow> 'b" shows "bounded_linear f \<equiv> linear f \<and> bounded_linear_axioms f"
enum_fin_string_def: shows "enum_class.enum \<equiv> fin_string_enum"
bind_option_K: fixes x :: "'c option"   and m :: "'d"   and fail :: "'d" assumes "x = None \<Longrightarrow> m = fail" shows "monad.bind_option fail x (\<lambda>_. m) = m"
meta_le_everything_if_top: fixes m :: "'c"   and x :: "'c" assumes "m = top" shows "x \<le> m"
simps_3: fixes t :: "trm"   and T :: "ty"   and y :: "vrs"   and t' :: "trm" shows "(t \<cdot>\<^sub>\<tau> T)[y \<mapsto> t'] = t[y \<mapsto> t'] \<cdot>\<^sub>\<tau> T"
interrupts_conv: fixes ls :: "'b"   and ts :: "'c"   and m :: "'d"   and ws :: "'e"   and "is" :: "'a" shows "interrupts (ls, (ts, m), ws, is) = is"
h1: shows "\<forall>n<dim_vec input_vec. input_vec $ n \<noteq> 0 \<longrightarrow> List.member (find_nonzeros_from_input_vec input_vec) n"
CallUndefined: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and p :: "'b"   and s :: "'a" assumes "\<Gamma> p = None" shows "\<Gamma>\<turnstile> (Call p, Normal s) \<rightarrow> (Skip, Stuck)"
rename_guarantees_eq_rename_inv: fixes h :: "'c \<Rightarrow> 'd"   and F :: "'c program"   and X :: "'d program set"   and Y :: "'d program set" assumes "bij h" shows "(rename h F \<in> X guarantees Y) = (F \<in> rename (inv h) ` X guarantees rename (inv h) ` Y)"
tbisim_def: fixes bisim :: "'g \<Rightarrow> 'h \<times> 'i \<Rightarrow> 'j \<times> 'k \<Rightarrow> bool"   and bisim_wait :: "'h \<Rightarrow> 'j \<Rightarrow> bool"   and nw :: "bool"   and t :: "'g"   and ts1 :: "('h \<times> 'l \<Rightarrow>f nat) option"   and m1 :: "'i"   and ts2 :: "('j \<times> 'l \<Rightarrow>f nat) option"   and m2 :: "'k" shows "FWbisimulation_base.tbisim bisim bisim_wait nw t ts1 m1 ts2 m2 = (case ts1 of None \<Rightarrow> ts2 = None | \<lfloor>(x1, ln)\<rfloor> \<Rightarrow> \<exists>x2. ts2 = \<lfloor>(x2, ln)\<rfloor> \<and> bisim t (x1, m1) (x2, m2) \<and> (nw \<or> bisim_wait x1 x2))"
induct: fixes P :: "nat \<Rightarrow> nat \<Rightarrow> bool"   and a0 :: "nat"   and a1 :: "nat" assumes "\<And>k n. \<lbrakk>\<lbrakk>n = 0; k \<noteq> 0\<rbrakk> \<Longrightarrow> P (k - 1) n; n \<noteq> 0 \<Longrightarrow> P (k - 1) (n div 2)\<rbrakk> \<Longrightarrow> P k n" shows "P a0 a1"
pick_injective_good: fixes X :: "('i, 'j, 'varSort, 'var, 'k) qTerm set"   and Y :: "('i, 'j, 'varSort, 'var, 'k) qTerm set" assumes "good X"   and "good Y" shows "(pick X = pick Y) = (X = Y)"
closed_halfspace_Im_le: fixes b :: "real" shows "closed {z. Im z \<le> b}"
Cod_cod: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "Cod (cod f) = Cod f"
lim: shows "LIM x at_right 0. f' x / g' x :> F"
gen_equals_def: fixes ball :: "'f \<Rightarrow> ('g \<times> 'h \<Rightarrow> bool) \<Rightarrow> bool"   and lu :: "'g \<Rightarrow> 'f \<Rightarrow> 'i option"   and eq :: "'i \<Rightarrow> 'h \<Rightarrow> bool"   and f :: "'f"   and g :: "'f" shows "gen_equals ball lu eq f g \<equiv> ball f (\<lambda>(k, v). rel_option eq (lu k g) (Some v)) \<and> ball g (\<lambda>(k, v). rel_option eq (lu k f) (Some v))"
simps_4: fixes I :: "interp"   and f :: "char"   and \<theta> :: "trm" shows "term_sem I (trm.Func f \<theta>) = (\<lambda>\<omega>. Funcs I f (term_sem I \<theta> \<omega>))"
rk_in_range: shows "k mod p \<in> {0..k}"
sterms_choice_disj: fixes \<Gamma> :: "'b \<Rightarrow> ('c, 'd, 'b, 'e) seqp"   and p :: "('c, 'd, 'b, 'e) seqp"   and p1 :: "('c, 'd, 'b, 'e) seqp"   and p2 :: "('c, 'd, 'b, 'e) seqp" assumes "wellformed \<Gamma>" shows "(p \<in> sterms \<Gamma> (p1 \<oplus> p2)) = (p \<in> sterms \<Gamma> p1 \<or> p \<in> sterms \<Gamma> p2)"
wtSB_def: fixes \<pi> :: "Preliminaries.var \<Rightarrow> 'fsym trm" shows "wtSB \<pi> \<equiv> \<forall>x. wt (\<pi> x) \<and> tpOf (\<pi> x) = tpOfV x"
unfold_llist_eq_LCons: fixes IS_LNIL :: "'d \<Rightarrow> bool"   and LHD :: "'d \<Rightarrow> 'c"   and LTL :: "'d \<Rightarrow> 'd"   and b :: "'d"   and x :: "'c"   and xs :: "'c llist" shows "(unfold_llist IS_LNIL LHD LTL b = LCons x xs) = (\<not> IS_LNIL b \<and> x = LHD b \<and> xs = unfold_llist IS_LNIL LHD LTL (LTL b))"
length_pps_to_list: fixes S :: "'t set" shows "length (pps_to_list S) = card S"
dsr1: fixes x :: "'a" shows "x + d x \<cdot> x = d x \<cdot> x"
HomFtor'_def: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and X :: "'d" shows "HomP\<^bsub>C\<^esub>[X,\<midarrow>] \<equiv> \<lparr>CatDom = C, CatCod = SET, MapM = HomFtorMap C X\<rparr>"
alg_dvd_def: fixes x :: "'a"   and y :: "'a" shows "(x alg_dvd y) = ((x = (0::'a) \<longrightarrow> y = (0::'a)) \<and> algebraic_int (y / x))"
Map_comp': fixes t' :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and t :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.seq t' t" shows "J_C.Map (t' \<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>] t) = vertical_composite.map (\<cdot>\<^sub>J) (\<cdot>) (J_C.Map t) (J_C.Map t')"
assms_1: shows "\<langle>x\<rangle> \<Turnstile>\<^sub>c\<^sub>s set (from_index_geq0_vector i v)"
findr_def: fixes e :: "nat" shows "findr e \<equiv> if \<exists>i<e_length e. e_nth e i \<noteq> 0 then Some (GREATEST i. i < e_length e \<and> e_nth e i \<noteq> 0) else Some (e_length e)"
degree_poly_of_vec_less: fixes v :: "'a vec" assumes "0 < dim_vec v" shows "degree (poly_of_vec v) < dim_vec v"
maddux_23: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x ; y \<cdot> - (x ; z) = x ; (y \<cdot> - z) \<cdot> - (x ; z)"
gauss_coprime_separable: fixes k :: "nat" assumes "coprime k n" shows "separable k"
assms: shows "<\<nu>x>\<tau>.P \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> P'"
unit_in_hom_2: fixes a :: "'b" assumes "B.obj a" shows "\<guillemotleft>\<tau>.G.unit a : \<tau>.G.map\<^sub>0 a \<Rightarrow>\<^sub>C G a\<guillemotright>"
recursive_compile_correct_norm': fixes f :: "recf"   and ap :: "abc_inst list"   and arity :: "nat"   and ft :: "nat"   and args :: "nat list" assumes "rec_ci f = (ap, arity, ft)"   and "terminate f args" shows "\<exists>stp rl. abc_steps_l (0, args) ap stp = (length ap, rl) \<and> abc_list_crsp (args @ [rec_exec f args]) rl"
l_def: shows "l \<equiv> liminf (\<lambda>n. ereal (f n / f (Suc n)))"
lebesgue_measure_ball_pos_2: shows "0 < content (cball (0::'b) 1)"
bound_on_all_plans_bounds_problem_plan_bound: fixes P :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> bool"   and f :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) set \<Rightarrow> nat" assumes "\<forall>PROB as s. P PROB \<and> finite PROB \<and> s \<in> valid_states PROB \<and> as \<in> valid_plans PROB \<longrightarrow> (\<exists>as'. exec_plan s as = exec_plan s as' \<and> subseq as' as \<and> length as' < f PROB)" shows "\<forall>PROB. P PROB \<and> finite PROB \<longrightarrow> problem_plan_bound PROB < f PROB"
funion_fminus_cancel: fixes A :: "'b fset"   and B :: "'b fset" shows "A |\<union>| (B |-| A) = A |\<union>| B"
simps: fixes x :: "'c"   and qs :: "'c list"   and i :: "nat"   and s :: "'c list \<times> 'd" shows "ALG x qs i s = (if x < qs ! i in fst s then 1 else 0)"
i'_bot: shows "i * 2 ^ (l' - l) < i'"
dirichlet_prod_const_right: fixes n :: "nat"   and f :: "nat \<Rightarrow> 'b"   and c :: "'b" assumes "0 < n" shows "dirichlet_prod f (\<lambda>n. if n = Suc 0 then c else (0::'b)) n = f n * c"
f': shows "\<forall>\<^sub>F x in F. 0 \<le> f x"
ex_orthogonal_current: shows "\<exists>f S. web_flow (web_of_network \<Delta>) f \<and> separating (web_of_network \<Delta>) S \<and> orthogonal_current (web_of_network \<Delta>) f S"
IfFalse: fixes b :: "bexp"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and \<pi> :: "char list \<Rightarrow> com option"   and c\<^sub>2 :: "com"   and t :: "char list \<Rightarrow> int \<Rightarrow> int"   and c\<^sub>1 :: "com" assumes "\<not> bval b s"   and "\<pi>: (c\<^sub>2, s) \<Rightarrow> t" shows "\<pi>: (IF b THEN c\<^sub>1 ELSE c\<^sub>2, s) \<Rightarrow> t"
intros_4: fixes \<gamma> :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and m :: "'c match_expr"   and p :: "'d"   and \<Gamma> :: "char list \<Rightarrow> 'c rule list option" assumes "matches \<gamma> m p" shows "\<Gamma>,\<gamma>,p\<turnstile> \<langle>[Rule m Reject], Undecided\<rangle> \<Rightarrow> Decision FinalDeny"
weakSimMonotonic: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and A :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set"   and Q :: "('a, 'b, 'c) psi"   and B :: "('b \<times> ('a, 'b, 'c) psi \<times> ('a, 'b, 'c) psi) set" assumes "\<Psi> \<rhd> P \<leadsto><A> Q"   and "A \<subseteq> B" shows "\<Psi> \<rhd> P \<leadsto><B> Q"
assms_1: shows "t + s \<in> existence_ivl t0 x0"
d_equal_trans: fixes DA :: "'a \<Rightarrow> 'b"   and d :: "'b"   and m :: "'a \<Rightarrow> 'c"   and m' :: "'a \<Rightarrow> 'c"   and m'' :: "'a \<Rightarrow> 'c" assumes "Strong_Security.d_equal DA d m m'"   and "Strong_Security.d_equal DA d m' m''" shows "Strong_Security.d_equal DA d m m''"
d_indistinguishable_def: fixes d :: "'d"   and e1 :: "'exp"   and e2 :: "'exp" shows "e1 \<equiv>\<^bsub>d\<^esub> e2 \<equiv> \<forall>m m'. m =\<^bsub>d\<^esub> m' \<longrightarrow> E e1 m = E e2 m'"
is_OK_returns_result_I: fixes h :: "'m"   and f :: "('m, 'n, 'o) prog"   and y :: "'o" assumes "h \<turnstile> f \<rightarrow>\<^sub>r y" shows "h \<turnstile> ok f"
inner_ge_zero: fixes x :: "'a" shows "0 \<le> inner x x"
max_Var_floatariths_rk2_fas: fixes x0 :: "floatarith list"   and cx :: "floatarith list"   and rkp :: "floatarith"   and h :: "floatarith"   and s2 :: "floatarith"   and z :: "nat" assumes "0 < length ode_e"   and "max_Var_floatariths ode_e \<le> length ode_e"   and "length x0 = length ode_e"   and "length cx = length ode_e"   and "Max {max_Var_floatarith rkp, max_Var_floatariths x0, max_Var_floatarith h, max_Var_floatariths cx, max_Var_floatarith s2} \<le> z" shows "max_Var_floatariths (rk2_fas rkp x0 h cx s2) \<le> z"
Red_I_to_Inf: fixes N :: "'a set" shows "Red_I N \<subseteq> Inf"
prems_3: shows "cs\<^bsup>\<pi>\<^esup> n = cs\<^bsup>\<pi>\<^esup> n'"
before_in_index2: fixes l :: "'c list"   and x :: "'c"   and y :: "'c" assumes "set l = {x, y}"   and "length l = 2"   and "x \<noteq> y" shows "(if x < y in l then 1 else 0) = index l y"
case_one_imp_not_learn_lim_V: fixes i :: "nat" assumes "case_one i" shows "\<not> learn_lim \<phi> (V_bclim i) (\<sigma> i)"
is_Var_q_Var: fixes i :: "name" shows "is_Var (q_Var i)"
rel_induct: fixes R :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x :: "'c list"   and y :: "'d list"   and Q :: "'c list \<Rightarrow> 'd list \<Rightarrow> bool" assumes "list_all2 R x y"   and "Q [] []"   and "\<And>a21 a22 b21 b22. \<lbrakk>R a21 b21; Q a22 b22\<rbrakk> \<Longrightarrow> Q (a21 # a22) (b21 # b22)" shows "Q x y"
cinner_zero_right: fixes scaleR :: "real \<Rightarrow> 'a \<Rightarrow> 'a"   and scaleC :: "complex \<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 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 cinner :: "'a \<Rightarrow> 'a \<Rightarrow> complex"   and x :: "'a" assumes "class.complex_inner scaleR scaleC plus zero minus uminus dist norm sgn uniformity open cinner" shows "cinner x zero = 0"
nonzero_iff: fixes xs :: "'a list" shows "nonzero xs = (\<exists>x\<in>set xs. x \<noteq> (0::'a))"
remove_subset: fixes x :: "'a"   and S :: "'a set" assumes "x \<in> S" shows "S - {x} \<subset> S"
bisim1Sync12: fixes P :: "'b prog"   and h :: "'heap"   and V :: "nat"   and e1 :: "'addr expr1"   and e2 :: "'addr expr1"   and a :: "'addr"   and xs :: "'addr val list"   and v :: "'addr val"   and v' :: "'addr val" shows "P,sync\<^bsub>V\<^esub> (e1) e2,h \<turnstile> (Throw a, xs) \<leftrightarrow> ([v, v'], xs, 4 + length (compE2 e1) + length (compE2 e2), \<lfloor>a\<rfloor>)"
knows_Notes: fixes A :: "agent"   and X :: "msg"   and evs :: "event list" shows "knows A (Notes A X # evs) = insert X (knows A evs)"
im_set_un2: fixes A :: "'c set"   and A1 :: "'c set"   and A2 :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "A = A1 \<union> A2" shows "f ` A = f ` A1 \<union> f ` A2"
to_poly_minus: fixes a :: "'a"   and b :: "'a" assumes "a \<in> carrier R"   and "b \<in> carrier R" shows "to_polynomial R (a \<ominus> b) = to_polynomial R a \<ominus>\<^bsub>P\<^esub> to_polynomial R b"
inf_absorb: fixes x :: "'a"   and y :: "'a" shows "x \<sqinter> (x \<squnion> y) = x"
inject_9: fixes x111 :: "('d, 'e, 'f) com"   and x112 :: "('d, 'e, 'f) com"   and y111 :: "('d, 'e, 'f) com"   and y112 :: "('d, 'e, 'f) com" shows "(Catch x111 x112 = Catch y111 y112) = (x111 = y111 \<and> x112 = y112)"
simps_4: fixes r :: "'b rexp"   and s :: "'b rexp" shows "\<guillemotleft>Plus r s\<guillemotright> = flatten PLUS (toplevel_summands (Plus \<guillemotleft>r\<guillemotright> \<guillemotleft>s\<guillemotright>))"
eval_fds_newman_conv_infsetsum_2: fixes s :: "complex" assumes "1 < Re s" shows "(\<lambda>p. complex_of_real (ln (real p) / real p) * hurwitz_zeta (real p) s) abs_summable_on {p. prime p}"
exi_I: fixes H :: "fmla set"   and A :: "fmla"   and i :: "name"   and x :: "trm" assumes "H \<turnstile> A(i::=x)" shows "H \<turnstile> exi i A"
partition_def: fixes cmp :: "'b comparator"   and pivot :: "'b"   and xs :: "'b list" shows "local.partition cmp pivot xs = (filter (\<lambda>x. compare cmp x pivot = Less) xs, filter (\<lambda>y. compare cmp pivot y = Equiv) xs, filter (\<lambda>x. compare cmp x pivot = Greater) xs)"
dghm_proj_fst_def: fixes \<AA> :: "V"   and \<BB> :: "V" shows "\<pi>\<^sub>D\<^sub>G\<^sub>.\<^sub>1 \<AA> \<BB> = \<pi>\<^sub>D\<^sub>G (2\<^sub>\<nat>) (\<lambda>i. if i = []\<^sub>\<circ> \<Rightarrow> \<AA> | otherwise \<Rightarrow> \<BB>) []\<^sub>\<circ>"
brnL_unique: fixes cl :: "('test, 'atom, 'choice) cmd list"   and n1 :: "nat"   and j1 :: "nat"   and n2 :: "nat"   and j2 :: "nat" assumes "properL cl"   and "n1 < length cl \<and> j1 < brn (cl ! n1)"   and "n2 < length cl \<and> j2 < brn (cl ! n2)"   and "brnL cl n1 + j1 = brnL cl n2 + j2" shows "n1 = n2 \<and> j1 = j2"
gallery_double_cross_not_minimal1: fixes X :: "'b set set"   and f :: "'b \<Rightarrow> 'b"   and B :: "'b set"   and C :: "'b set"   and D :: "'b set"   and Bs :: "'b set list"   and Cs :: "'b set list" assumes "ThinChamberComplexFolding X f"   and "B \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "C \<in> ChamberComplex.chamber_system X - f \<turnstile> ChamberComplex.chamber_system X"   and "D \<in> f \<turnstile> ChamberComplex.chamber_system X"   and "SimplicialComplex.maxsimpchain X (B # Bs @ C # Cs @ [D])" shows "\<not> SimplicialComplex.min_maxsimpchain X (B # Bs @ C # Cs @ [D])"
set_cases: fixes e :: "'c"   and a :: "'c dual" assumes "e \<in> set_dual a" obtains "a = dual e"
interval_neqs_8: fixes r :: "real" shows "{} \<noteq> {..<r}"
l2_inv7E: fixes x :: "l2_state" assumes "x \<in> l2_inv7" obtains "extr (bad x) (ik x) (chan x) \<subseteq> synth (analz generators)"
even_diff_iff: fixes k :: "int"   and l :: "int" shows "even (k - l) = even (k + l)"
assms: fixes P :: "'a" assumes "P \<in> A" shows "F P = G P"
simps: fixes p :: "(action \<times> nat) list"   and e :: "nat" shows "adjust p e = map (\<lambda>(a, s). (a, if s = 0 then e else s)) p"
pow_le_max: fixes tid :: "'nid"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat" assumes "honest tid"   and "\<parallel>tid\<parallel>\<^bsub>t n\<^esub>" shows "PoW t n \<le> length (MAX (pin (\<sigma>\<^bsub>tid\<^esub>t n)))"
subset: shows "set (map (\<lambda>i. f (Suc i)) [0..<l]) \<subseteq> snd ` set rel"
del_arc_in: fixes a :: "'c"   and G :: "('d, 'c) pre_digraph" assumes "a \<notin> arcs G" shows "pre_digraph.del_arc G a = G"
oppOrdering_compare: fixes eq :: "'a \<rightarrow> 'a \<rightarrow> tr"   and compare :: "'a \<rightarrow> 'a \<rightarrow> Ordering"   and x :: "'a"   and y :: "'a" assumes "class.Ord_linear eq compare" shows "oppOrdering\<cdot>(compare\<cdot>x\<cdot>y) = compare\<cdot>y\<cdot>x"
tm_append_second_steps_eq: fixes s :: "nat"   and l :: "cell list"   and r :: "cell list"   and B :: "(action \<times> nat) list"   and stp :: "nat"   and s' :: "nat"   and l' :: "cell list"   and r' :: "cell list"   and off :: "nat"   and A :: "(action \<times> nat) list" assumes "steps0 (s, l, r) B stp = (s', l', r')"   and "s' \<noteq> 0"   and "off = length A div 2"   and "is_even (length A)" shows "steps0 (s + off, l, r) (A @ shift B off) stp = (s' + off, l', r')"
Cons_1: fixes L' :: "atomUni list" assumes "convert_atom_list var L (xs' @ x # xs) = Some L'"   and "length xs' = var" shows "(\<forall>f\<in>set L. eval (substNegInfinity var f) (xs' @ x # xs)) = (\<forall>f\<in>set L'. evalUni (substNegInfinityUni f) x)"
dec_unclocked: fixes mn :: "char list list"   and env :: "v sem_env"   and s :: "'b state"   and d :: "dec"   and s' :: "'b state"   and r :: "(v sem_env, v) result"   and count1 :: "nat"   and count2 :: "nat" shows "(evaluate_dec False mn env s d (s', r) \<longrightarrow> r \<noteq> Rerr (Rabort Rtimeout_error) \<and> clock s = clock s') \<and> (evaluate_dec False mn env (update_clock (\<lambda>_. count1) s) d (update_clock (\<lambda>_. count1) s', r) \<longrightarrow> evaluate_dec False mn env (update_clock (\<lambda>_. count2) s) d (update_clock (\<lambda>_. count2) s', r))"
assms_2: shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<AA>"