hcju/isabelleps huggingface dataset:
simps_2: fixes Ds :: "'a literal list list"   and C :: "'a literal list"   and L :: "'a literal"   and C' :: "'a literal list" shows "reduce Ds C (L # C') = (if is_reducible_lit Ds (C @ C') L then reduce Ds C C' else L # reduce Ds (L # C) C')"
productF_path_io_targets_reverse: fixes A :: "('d, 'e, 'f) FSM"   and B :: "('d, 'e, 'g) FSM"   and FAIL :: "'f \<times> 'g"   and AB :: "('d, 'e, 'f \<times> 'g) FSM"   and pA :: "'f"   and qA :: "'f"   and w :: "('d \<times> 'e) list"   and pB :: "'g"   and qB :: "'g" assumes "productF A B FAIL AB"   and "pA \<in> io_targets A qA w"   and "pB \<in> io_targets B qB w"   and "w \<in> LS A qA"   and "w \<in> LS B qB"   and "observable A"   and "observable B"   and "well_formed A"   and "well_formed B"   and "qA \<in> nodes A"   and "qB \<in> nodes B" shows "io_targets AB (qA, qB) w = {(pA, pB)}"
infinite: fixes A :: "'c set" assumes "infinite A" shows "Max A = the None"
ProcBody: fixes P :: "'d set"   and P' :: "'d set"   and \<Gamma> :: "'e \<Rightarrow> ('d, 'e, 'f) com option"   and \<Theta> :: "('d set \<times> 'e \<times> 'd set \<times> 'd set) set"   and F :: "'f set"   and body :: "('d, 'e, 'f) com"   and Q :: "'d set"   and A :: "'d set"   and p :: "'e" assumes "P \<subseteq> P'"   and "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P' body Q,A"   and "\<Gamma> p = Some body" shows "\<Gamma>,\<Theta>\<turnstile>\<^bsub>/F \<^esub>P Call p Q,A"
simps_1: fixes l :: "tree"   and r :: "tree" shows "internal [] (Branching l r) = True"
bisimE_1: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and Q :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> P \<sim> Q" shows "insertAssertion (extractFrame P) \<Psi> \<simeq>\<^sub>F insertAssertion (extractFrame Q) \<Psi>"
lfp_def: fixes c :: "com"   and f :: "'a acom \<Rightarrow> 'a acom" shows "Collecting.lfp c f = Inf_acom c {a \<in> {C. strip C = c}. f a \<le> a}"
arr_cod_iff_arr: fixes f :: "'a1 \<times> 'a2" shows "A1xA2.arr (A1xA2.cod f) = A1xA2.arr f"
lsorted_lSup: fixes Y :: "'a llist set" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) Y"   and "\<forall>xs\<in>Y. lsorted xs" shows "lsorted (lSup Y)"
norm_sq_mtx_def: fixes A :: "'b sq_mtx" shows "\<parallel>A\<parallel> = \<parallel>to_vec A\<parallel>\<^sub>o\<^sub>p"
Vset_rank_lt: fixes a :: "hf"   and k :: "hf" assumes "a \<^bold>\<in> Vset k" shows "rank a < k"
hyps: fixes v :: "'v" assumes "finite E"   and "is_trail v xs v'"   and "n \<notin> {v, v'}" shows "even (degree n (rem_unPath xs G)) = even (degree n G)"
continuous_imp_open_vimage: fixes s :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and B :: "'c set" assumes "continuous_on s f"   and "open s"   and "open B"   and "f -` B \<subseteq> s" shows "open (f -` B)"
eqx: shows "fst \<gamma>1' = fst \<gamma>1"
istopology_base: fixes P :: "'d set \<Rightarrow> bool" assumes "\<And>S T. \<lbrakk>P S; P T\<rbrakk> \<Longrightarrow> P (S \<inter> T)" shows "istopology (arbitrary union_of P)"
hyps_1: fixes D :: "(nat \<times> symbol \<times> symbol list) list"   and a :: "symbol list"   and u :: "symbol list"   and v :: "symbol list" assumes "n = length D"   and "Derivation a D (u @ v)"   and "is_word u" shows "\<exists>w. leftderives a (u @ w) \<and> (v = [] \<longrightarrow> w = []) \<and> (\<forall>X. is_first X v \<longrightarrow> is_first X w)"
step: fixes conv :: "('a, 'b, 'c, 'd) converter"   and a :: "'a" assumes "X conv"   and "a \<in> outs_\<I> \<I>" shows "colossless_gpv \<I>' (run_converter conv a) \<and> (\<forall>(b, conv')\<in>results_gpv \<I>' (run_converter conv a). X conv' \<or> colossless_converter \<I> \<I>' conv')"
inJ: fixes i :: "'c" assumes "i \<in> I" shows "S i \<subseteq> J"
bigstepT_the_state: fixes c :: "com"   and s :: "char list \<Rightarrow> int"   and P :: "nat"   and T :: "char list \<Rightarrow> int" assumes "(c, s) \<Rightarrow> P \<Down> T" shows "\<down>\<^sub>s (c, s) = T"
prem: shows "\<turnstile> F \<longrightarrow> G"
Chi_def: fixes A :: "('d, 'e, 'f) hierauto" shows "Chi A \<equiv> \<lambda>S\<in>HAStates A. {S'. \<exists>SA\<in>SAs A. SA \<in> the (CompFun A S) \<and> S' \<in> States SA}"
Int_lower1: fixes A :: "'b set"   and B :: "'b set" shows "order.greater_eq A (A \<inter> B)"
listrel_eq_len: fixes xs :: "'b list"   and ys :: "'c list"   and r :: "('b \<times> 'c) set" assumes "(xs, ys) \<in> listrel r" shows "length xs = length ys"
Bfun_def: fixes f :: "'b \<Rightarrow> 'c"   and F :: "'b filter" shows "Bfun f F = (\<exists>K>0. \<forall>\<^sub>F x in F. norm (f x) \<le> K)"
orientable_5_6: fixes x :: "'a" shows "(local.comp_inf.symmetric x \<longrightarrow> bounded_distrib_allegory.loop_orientable top (\<sqinter>) (\<le>) (\<squnion>) id x) = (local.comp_inf.symmetric x \<longrightarrow> (\<exists>y. y \<squnion> id y = x \<and> y \<sqinter> id y \<le> x \<sqinter> top))"
NOU2: shows "Oadmit \<sigma> ODE2 (BVO (OProd ODE1 ODE2))"
surjective_def: fixes f :: "'a \<Rightarrow> 'b" shows "surjective f \<equiv> surj f"
S2: shows "list_to_nat ls = c_pair (length ls - 1) (c_fold ls) + 1"
leg0_composite: shows "H\<mu>H\<nu>\<pi>.leg0 = \<nu>\<pi>.leg0 \<cdot> \<mu>_\<nu>\<pi>.prj\<^sub>0"
semicontraction_iterates: fixes f :: "'b \<Rightarrow> 'b"   and n :: "nat" assumes "1-lipschitz_on UNIV f" shows "1-lipschitz_on UNIV (f ^^ n)"
pr_pri_agrk_analz_intro: fixes evs :: "event list"   and S :: "agent \<times> nat \<times> nat \<Rightarrow> session"   and A :: "msg set"   and U :: "msg set"   and x :: "rat" assumes "(evs, S, A, U) \<in> protocol"   and "Pri_AgrK x \<in> analz (A \<union> spies evs)" shows "Pri_AgrK x \<in> A"
Class_equivalence: fixes a :: "'a"   and b :: "'a" assumes "a \<in> R"   and "b \<in> R" shows "(additive.Class a = additive.Class b) = ((a, b) \<in> Ring_Congruence)"
empty_subsetI: fixes A :: "'k set" shows "{} \<subseteq> A"
opar_bigger_than_next: fixes i :: "nat" shows "optoy i \<langle>\<langle>\<^bsub>i\<^esub> qmsg \<Turnstile> (otherwith nos_inc {i} (orecvmsg msg_ok), other nos_inc {i} \<rightarrow>) global (\<lambda>\<sigma>. no (\<sigma> i) \<le> no (\<sigma> (nhid (\<sigma> i))))"
base_pos: fixes base :: "nat" assumes "digits base" shows "0 < base"
assms_2: shows "bounded (err_step (length ts) app step) (length ts)"
closed\<^sub>T1: shows "T\<^sub>1 closed_in (TVarB X T\<^sub>2 # \<Gamma>)"
s2: shows "s2 = (ls2, (ts2, m2), ws2, is2)"
ltl_const_equiv_or_const_2: fixes \<phi>\<^sub>1 :: "'a ltln"   and \<phi>\<^sub>2 :: "'a ltln" shows "\<phi>\<^sub>1 or\<^sub>n \<phi>\<^sub>2 \<sim>\<^sub>C false\<^sub>n = (\<phi>\<^sub>1 \<sim>\<^sub>C false\<^sub>n \<and> \<phi>\<^sub>2 \<sim>\<^sub>C false\<^sub>n)"
exponent_le: fixes a :: "('a, 'b) IEEE.float" shows "IEEE.exponent a \<le> unat (- 1)"
swap_cols_rows_block_dims_1: fixes i :: "nat"   and j :: "nat"   and A :: "'b mat" shows "dim_row (??.Jordan_Normal_Form_Existence.swap_cols_rows_block i j A) = dim_row A"
simps_16: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a"   and f4 :: "'a" shows "(case T_f64 of T_i32 \<Rightarrow> f1 | T_i64 \<Rightarrow> f2 | T_f32 \<Rightarrow> f3 | T_f64 \<Rightarrow> f4) = f4"
bisimCoinduct: fixes P :: "ccs"   and Q :: "ccs"   and X :: "(ccs \<times> ccs) set" assumes "(P, Q) \<in> X"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> R \<leadsto>[(X \<union> bisim)] S"   and "\<And>R S. (R, S) \<in> X \<Longrightarrow> (S, R) \<in> X" shows "P \<sim> Q"
sum_mono: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and K :: "'d set"   and f :: "'d \<Rightarrow> 'c"   and g :: "'d \<Rightarrow> 'c" assumes "class.ordered_comm_monoid_add plus zero less_eq less"   and "\<And>i. i \<in> K \<Longrightarrow> less_eq (f i) (g i)" shows "less_eq (comm_monoid_add.sum plus zero f K) (comm_monoid_add.sum plus zero g K)"
a_de_morgan_var_4: fixes x :: "'a"   and y :: "'a" shows "ad (d x \<cdot> d y) = ad x + ad y"
map_iterator_product_def: fixes it_a :: "('d \<Rightarrow> bool) \<Rightarrow> ('e \<times> 'g \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd"   and it_b :: "'g \<Rightarrow> ('d \<Rightarrow> bool) \<Rightarrow> ('f \<Rightarrow> 'd \<Rightarrow> 'd) \<Rightarrow> 'd \<Rightarrow> 'd" shows "map_iterator_product it_a it_b = set_iterator_image (\<lambda>kvv'. (fst (fst kvv'), snd kvv')) (set_iterator_product it_a (\<lambda>kv. it_b (snd kv)))"
quantity_adapted: fixes p :: "'c \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat"   and Mkt :: "('a, 'c) discrete_market" assumes "\<forall>asset\<in>support_set p. p asset (Suc n) \<in> borel_measurable (F n)"   and "\<forall>asset\<in>support_set p. prices Mkt asset n \<in> borel_measurable (F n)" shows "val_process Mkt p n \<in> borel_measurable (F n)"
red_induct: fixes x1a :: "com"   and x1b :: "char list \<Rightarrow> val option"   and x2a :: "com"   and x2b :: "char list \<Rightarrow> val option"   and P :: "com \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> com \<Rightarrow> (char list \<Rightarrow> val option) \<Rightarrow> bool" assumes "\<langle>x1a,x1b\<rangle> \<rightarrow> \<langle>x2a,x2b\<rangle>"   and "\<And>V e s. P (V:=e) s Skip (s(V := \<lbrakk>e\<rbrakk>s))"   and "\<And>c\<^sub>1 s c\<^sub>1' s' c\<^sub>2. \<lbrakk>\<langle>c\<^sub>1,s\<rangle> \<rightarrow> \<langle>c\<^sub>1',s'\<rangle>; P c\<^sub>1 s c\<^sub>1' s'\<rbrakk> \<Longrightarrow> P (c\<^sub>1;; c\<^sub>2) s (c\<^sub>1';; c\<^sub>2) s'"   and "\<And>c\<^sub>2 s. P (Skip;; c\<^sub>2) s c\<^sub>2 s"   and "\<And>b s c\<^sub>1 c\<^sub>2. \<lbrakk>b\<rbrakk>s = Some true \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) s c\<^sub>1 s"   and "\<And>b s c\<^sub>1 c\<^sub>2. \<lbrakk>b\<rbrakk>s = Some false \<Longrightarrow> P (if (b) c\<^sub>1 else c\<^sub>2) s c\<^sub>2 s"   and "\<And>b s c. \<lbrakk>b\<rbrakk>s = Some true \<Longrightarrow> P (while (b) c) s (c;; while (b) c) s"   and "\<And>b s c. \<lbrakk>b\<rbrakk>s = Some false \<Longrightarrow> P (while (b) c) s Skip s" shows "P x1a x1b x2a x2b"
implied_frontier_alt_def: fixes c :: "('loc, 't, 'a) configuration_scheme"   and loc :: "'loc" shows "implied_frontier_alt c loc = frontier (\<Sum>loc'\<in>UNIV. after_summary (zmset_of (mset_set (set_antichain (frontier (c_pts c loc'))))) (path_summary loc' loc))"
set_fmap_pair: fixes x :: "'g \<times> 'h"   and F :: "'g \<Rightarrow> 'h option" assumes "x \<in> set_fmap F" shows "fst x \<in> dom F \<and> snd x = the (F (fst x))"
lit_fun_simps_4: fixes f :: "'e \<Rightarrow> 'c"   and x :: "'e" shows "\<guillemotleft>f x\<guillemotright> = uop f \<guillemotleft>x\<guillemotright>"
boundary_def: fixes twoC :: "real \<times> real \<Rightarrow> real \<times> real" shows "boundary twoC \<equiv> horizontal_boundary twoC \<union> vertical_boundary twoC"
insert_not_empty: fixes a :: "'b::type"   and A :: "'b::type set" shows "insert (a::'b::type) (A::'b::type set) \<noteq> {}"
below_refl: fixes x :: "'a" shows "x \<sqsubseteq> x"
partition_on_insert': fixes A :: "'c set"   and X :: "'c set"   and P :: "'c set set" assumes "partition_on (A - X) P"   and "X \<subseteq> A"   and "X \<noteq> {}" shows "partition_on A (insert X P)"
assms: shows "a \<in>\<^sub>\<circ> \<rat>\<^sub>\<circ>"
enat_add_sub_same2: fixes b :: "enat"   and a :: "enat" assumes "b \<noteq> \<infinity>" shows "a + b - b = a"
value_f: shows "value_flow \<Delta> (\<lambda>x. ennreal (f' x)) = \<alpha>"
tdghm_dghm_comp_is_tdghm: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V"   and \<FF> :: "V"   and \<GG> :: "V"   and \<NN> :: "V"   and \<AA> :: "V"   and \<HH> :: "V" assumes "\<NN> : \<FF> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> : \<BB> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"   and "\<HH> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<NN> \<circ>\<^sub>T\<^sub>D\<^sub>G\<^sub>H\<^sub>M\<^sub>-\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH> : \<FF> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH> \<mapsto>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<GG> \<circ>\<^sub>D\<^sub>G\<^sub>H\<^sub>M \<HH> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
turn_into_distance_def: fixes f :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and x :: "'b"   and y :: "'b" shows "turn_into_distance f x y = Inf {\<Sum>i = 0..<n. f (u i) (u (Suc i)) |u n. u 0 = x \<and> u n = y}"
bd_cts_def: fixes t :: "('b \<Rightarrow> real) \<Rightarrow> 'b \<Rightarrow> real" shows "bd_cts t = (\<forall>M. (\<forall>i. M i \<tturnstile> M (Suc i) \<and> sound (M i)) \<longrightarrow> (\<exists>b. \<forall>i. bounded_by b (M i)) \<longrightarrow> t (Sup_exp (range M)) = Sup_exp (range (t \<circ> M)))"
inv_flip_bv: fixes i :: "nat"   and n :: "nat" shows "map_pmf (flip i) (bv n) = bv n"
convert_substInfinitesimalLinear_less: fixes var :: "nat"   and p :: "real mpoly"   and xs' :: "real list"   and x :: "real"   and xs :: "real list"   and p' :: "real \<times> real \<times> real"   and b :: "real mpoly"   and B :: "real"   and c :: "real mpoly"   and C :: "real" assumes "convert_poly var p (xs' @ x # xs) = Some p'"   and "insertion (nth_default 0 (xs' @ x # xs)) b = B"   and "insertion (nth_default 0 (xs' @ x # xs)) c = C"   and "B \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "length xs' = var" shows "eval (liftmap (\<lambda>x A. fm.Atom (linear_substitution (var + x) (liftPoly 0 x (- c)) (liftPoly 0 x b) A)) (convertDerivative var p) 0) (xs' @ x # xs) = evalUni (map_atomUni (linearSubstitutionUni B C) (convertDerivativeUni p')) x"
vset_disj_eff_diff: fixes a :: "('d, bool) fmap \<times> ('d, bool) fmap"   and vs :: "'d set"   and s :: "'d set" assumes "varset_action a vs" shows "fmdom' (snd a) \<inter> (s - vs) = {}"
eq: shows "(h \<star> \<theta>) \<cdot> \<a>[h, p\<^sub>0, w] \<cdot> (\<phi> \<star> w) \<cdot> \<a>\<^sup>-\<^sup>1[k, p\<^sub>1, w] = (h \<star> \<theta>') \<cdot> \<a>[h, p\<^sub>0, w'] \<cdot> (\<phi> \<star> w') \<cdot> \<a>\<^sup>-\<^sup>1[k, p\<^sub>1, w'] \<cdot> (k \<star> \<beta>)"
conjug_eq_prim_rootE: fixes x :: "'b list"   and z :: "'b list"   and y :: "'b list" assumes "x \<cdot> z = z \<cdot> y"   and "x \<noteq> \<epsilon>" obtains r :: "'b list" and s :: "'b list" and i :: "nat" and n :: "nat" where "(r \<cdot> s) \<^sup>@ Suc i = x" and "(s \<cdot> r) \<^sup>@ Suc i = y" and "(r \<cdot> s) \<^sup>@ n \<cdot> r = z" and "s \<noteq> \<epsilon>" and "primitive (r \<cdot> s)"
eqvt_fresh_cong3: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> 'f"   and a :: "atom"   and x :: "'c"   and y :: "'d"   and z :: "'e" assumes "\<And>p x y z. p \<bullet> f x y z = f (p \<bullet> x) (p \<bullet> y) (p \<bullet> z)"   and "a \<sharp> x"   and "a \<sharp> y"   and "a \<sharp> z" shows "a \<sharp> f x y z"
AL_Size_UpdateSuc: fixes L :: "('b \<times> 'a) list"   and a :: "'b"   and b :: "'a" assumes "L\<down>a = None" shows "|L[a\<mapsto>b]| = Suc |L|"
mult_le_mono1: fixes i :: "nat"   and j :: "nat"   and k :: "nat" assumes "i \<le> j" shows "i * k \<le> j * k"
induct: fixes x :: "msg"   and H :: "msg set"   and P :: "msg \<Rightarrow> bool" assumes "x \<in> crypts H"   and "\<And>X. X \<in> H \<Longrightarrow> P X"   and "\<And>KS X. \<lbrakk>foldr Crypt KS (Hash X) \<in> crypts H; P (foldr Crypt KS (Hash X))\<rbrakk> \<Longrightarrow> P (foldr Crypt KS X)"   and "\<And>KS X Y. \<lbrakk>foldr Crypt KS \<lbrace>X, Y\<rbrace> \<in> crypts H; P (foldr Crypt KS \<lbrace>X, Y\<rbrace>)\<rbrakk> \<Longrightarrow> P (foldr Crypt KS X)"   and "\<And>KS X Y. \<lbrakk>foldr Crypt KS \<lbrace>X, Y\<rbrace> \<in> crypts H; P (foldr Crypt KS \<lbrace>X, Y\<rbrace>)\<rbrakk> \<Longrightarrow> P (foldr Crypt KS Y)" shows "P x"
assms: fixes x :: "'a" assumes "x \<in> set xs" shows "x \<noteq> (0::'a)"
per_pref': fixes u :: "'b list"   and v :: "'b list"   and k :: "nat" assumes "u \<noteq> \<epsilon>"   and "periodN v k"   and "u \<le>p v" shows "periodN u k"
ext_inject: fixes src :: "'a"   and tgt :: "'a"   and more :: "'b"   and src' :: "'a"   and tgt' :: "'a"   and more' :: "'b" shows "(\<lparr>src = src, tgt = tgt, \<dots> = more\<rparr> = \<lparr>src = src', tgt = tgt', \<dots> = more'\<rparr>) = (src = src' \<and> tgt = tgt' \<and> more = more')"
rel_interior_convex_hull_explicit: fixes S :: "'a set" assumes "\<not> affine_dependent S" shows "rel_interior (convex hull S) = {y. \<exists>u. (\<forall>x\<in>S. 0 < u x) \<and> sum u S = 1 \<and> (\<Sum>x\<in>S. u x *\<^sub>R x) = y}"
simps: fixes k :: "nat"   and v :: "'b"   and x :: "(nat \<times> 'b) list \<times> nat option list" shows "insert_idx_pqueue_fun k v x = (let x' = idx_pqueue_push_fun k v x in idx_bubble_up_fun x' (length (fst x') - 1))"
MakeCatId: fixes X :: "'a"   and C :: "('a, 'b, 'c) Category_scheme" assumes "X \<in> obj\<^bsub>C\<^esub>" shows "id\<^bsub>C\<^esub> X = id\<^bsub>MakeCat C\<^esub> X"
simps_2: fixes h :: "'b heap" shows "exec Del_min [h] = del_min h"
monoid_axioms: shows "monoid (\<^bold>\<sqinter>) \<one>"
inv2a: shows "Inv2a_innermost s p (dblock s p)"
arr_dom_iff_arr: fixes f :: "'a" shows "Left.arr (Left.dom f) = Left.arr f"
wf_since_aux_Nil: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and n :: "nat"   and R :: "'a list set"   and pos :: "bool"   and \<phi>' :: "'a MFOTL.formula"   and I :: "\<I>"   and \<psi>' :: "'a MFOTL.formula" shows "wf_since_aux \<sigma> n R pos \<phi>' I \<psi>' [] 0"
append_Nil: fixes ys :: "'a list" shows "append [] ys = ys"
edgeDecls_finite: fixes e :: "edge" shows "finite (edgeDecls e)"
arr_def: fixes f :: "'a" shows "arr f \<equiv> domains f \<noteq> {} \<or> codomains f \<noteq> {}"
insert_preserve_indices: fixes xs :: "('c \<times> 'd \<times> bool) list"   and e :: "'c \<times> 'd \<times> bool"   and i :: "'c option" assumes "\<exists>ys. insert xs e i = Some ys" shows "fst ` set (the (insert xs e i)) = fst ` set xs \<union> {fst e}"
pfsubset_imp_fsubset: fixes A :: "'b fset"   and B :: "'b fset" assumes "A |\<subset>| B" shows "A |\<subseteq>| B"
order_trans: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a"   and y :: "'a"   and z :: "'a" assumes "class.preorder less_eq less"   and "less_eq x y"   and "less_eq y z" shows "less_eq x z"
subset_transfer: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "bi_unique A" shows "rel_fun (rel_set A) (rel_fun (rel_set A) (=)) (\<subseteq>) (\<subseteq>)"
pairwise_def: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and M :: "'a set" shows "Compiler_Utils.pairwise P M = (\<forall>m\<in>M. \<forall>n\<in>M. P m n)"
ran_upd: fixes m :: "'b \<Rightarrow> 'a option"   and k :: "'b"   and v :: "'a" shows "ran (m(k \<mapsto> v)) \<subseteq> ran m \<union> {v}"
assms_1: shows "RSD_pareto_eqclass agents' alts R A"
degree_smult_eq_iff: fixes a :: "'b"   and p :: "'b poly" shows "(degree (Polynomial.smult a p) = degree p) = (degree p = 0 \<or> a * lead_coeff p \<noteq> (0::'b))"
sets_negligible_symdiff: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> sets lebesgue"   and "negligible (S - T \<union> (T - S))" shows "T \<in> sets lebesgue"
no_zero_cycle: fixes loc :: "'loc"   and xs :: "('loc \<times> 'sum \<times> 'loc) list"   and s :: "'sum"   and t :: "'t" assumes "path loc loc xs"   and "xs \<noteq> []"   and "s = sum_weights (map (\<lambda>(s, l, t). l) xs)" shows "t < results_in t s"
thetaZOParT_ZOretr: shows "thetaZOParT \<subseteq> ZOretr thetaZOParT"
MInfty_eq_minfinity: shows "MInfty = - \<infinity>"
induced_ring_hom: fixes K :: "'a set" assumes "subring K R" shows "ring_hom_ring (R\<lparr>carrier := K\<rparr>) S h"
size_multiset_int_count: fixes A :: "'a multiset"   and ca :: "int"   and ca2 :: "int" assumes "int (card (set_mset A)) = ca"   and "\<And>p. p \<in># A \<Longrightarrow> int (count A p) = ca2" shows "int (size A) = ca * ca2"
sqr_conv_mult: fixes x :: "num" shows "Num.sqr x = x * x"
sat_imp: fixes p :: "'b pl"   and H :: "'b pl set"   and q :: "'b pl" assumes "insert p H |= q" shows "H |= p -> q"
statement66_existence: fixes a1 :: "proj2"   and a2 :: "proj2"   and p1 :: "proj2"   and p2 :: "proj2" assumes "a1 \<in> hyp2"   and "a2 \<in> hyp2"   and "p1 \<in> S"   and "p2 \<in> S" shows "\<exists>J. is_K2_isometry J \<and> apply_cltn2 a1 J = a2 \<and> apply_cltn2 p1 J = p2"
exec_instr_2: shows "(ta, xcp', h', [(stk', loc', undefined, undefined, pc')]) \<in> exec_instr (ins ! pc) P t h stk loc undefined undefined pc []"
\<iota>_def: shows "\<iota> = Infer (CAs @ [D]) E"
log2_def: fixes a :: "int" shows "log2 a = \<lfloor>log 2 (real_of_int a)\<rfloor>"
mn_LE_def: shows "mn_LE \<equiv> inv_image Recursive.lex_triple mn_measure"
Apps_alt: fixes f :: "expr"   and ts :: "expr list"   and T :: "Unary_PCF.type" shows "f \<bullet> ts ::: T = (\<exists>Ts. f ::: fold (\<rightarrow>) (rev Ts) T \<and> list_all2 (:::) ts Ts)"
homologous_rel_eq_relboundary: fixes p :: "nat"   and X :: "'b topology"   and S :: "'b set"   and c :: "((nat \<Rightarrow> real) \<Rightarrow> 'b) \<Rightarrow>\<^sub>0 int" shows "(homologous_rel p X S c = singular_relboundary p X S) = singular_relboundary p X S c"
less_le_not_le: fixes x :: "'b zmultiset"   and y :: "'b zmultiset" shows "(x \<subset>#\<^sub>z y) = (x \<subseteq>#\<^sub>z y \<and> \<not> y \<subseteq>#\<^sub>z x)"
det_def: fixes A :: "'c mat" shows "Determinant.det A = (if dim_row A = dim_col A then \<Sum>p | p permutes {0..<dim_row A}. signof p * (\<Prod>i = 0..<dim_row A. A $$ (i, p i)) else (0::'c))"
mems_le_finite_i: fixes s :: "nat set"   and k :: "nat" assumes "\<forall>x. x \<in> s \<longrightarrow> x \<le> k" shows "finite s"
generalizes_atm_def: fixes A :: "'a"   and B :: "'a" shows "generalizes_atm A B = (\<exists>\<sigma>. A \<cdot>a \<sigma> = B)"
fim: shows "f ` (sphere (0::'a) 1 \<inter> S) \<subseteq> sphere (0::'a) 1 \<inter> T"
cr_rw: shows "code_rate enc X = (\<Sum>i\<in>X ` space M. fi i * real (cw_len i))"
scaleR_nonneg_nonneg: fixes a :: "real"   and x :: "'b" assumes "0 \<le> a"   and "(0::'b) \<le> x" shows "(0::'b) \<le> a *\<^sub>R x"
quotient_of_denom_diff_dvd: fixes x :: "rat"   and y :: "rat" shows "snd (quotient_of (x - y)) dvd snd (quotient_of x) * snd (quotient_of y)"
is_vals_conv: fixes es :: "('a, 'b, 'c) exp list" shows "is_vals es = (\<exists>vs. es = map Val vs)"
index_zero_implies_same_group: fixes point_set :: "'b set"   and block_collection :: "'b set multiset"   and groups :: "'b set set"   and index :: "int"   and ps :: "'b set" assumes "GDD point_set block_collection groups index"   and "ps \<subseteq> point_set"   and "card ps = 2"   and "block_collection index ps = 0" shows "\<exists>G\<in>groups. ps \<subseteq> G"
sdom_asyc: fixes i :: "nat"   and j :: "nat" assumes "strict_dom i j"   and "j \<in> set (g_V G)" shows "i \<noteq> j"
f_def: shows "f \<equiv> \<lambda>u. \<Sum>j\<in>Basis. (u \<bullet> basg j) *\<^sub>R j"
Gr_not_in: fixes x :: "'b"   and F :: "'b set"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "x \<notin> F \<or> f x \<noteq> y" shows "(x, y) \<notin> BNF_Def.Gr F f"
x0: shows "x0 = \<lbrakk>\<guillemotleft>A(decode_Var v::=Zero)\<guillemotright>\<rbrakk>e"
div_by_1: 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 a :: "'a" assumes "class.semidom_divide divide plus minus zero times one" shows "divide a one = a"
type_wellformed_def: fixes ev\<^sub>A :: "('a \<Rightarrow> 'b) \<Rightarrow> 'c \<Rightarrow> 'b"   and ev\<^sub>B :: "('a \<Rightarrow> 'b) \<Rightarrow> 'd \<Rightarrow> bool"   and aexp_vars :: "'c \<Rightarrow> 'a set"   and bexp_vars :: "'d \<Rightarrow> 'a set"   and dma :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> Sec"   and \<C>_vars :: "'a \<Rightarrow> 'a set"   and bexp_neg :: "'d \<Rightarrow> 'd"   and assign_post :: "'d set \<Rightarrow> 'a \<Rightarrow> 'c \<Rightarrow> 'd set"   and dma_type :: "'a \<Rightarrow> 'd set"   and pred_False :: "'d"   and t :: "'d set" assumes "sifum_types ev\<^sub>A ev\<^sub>B aexp_vars bexp_vars dma \<C>_vars bexp_neg assign_post dma_type pred_False" shows "sifum_types.type_wellformed bexp_vars \<C>_vars t \<equiv> \<Union> (bexp_vars ` t) \<subseteq> \<Union> (range \<C>_vars)"