hcju/isabelleps huggingface dataset:
sum_gp: fixes x :: "'a"   and m :: "nat"   and n :: "nat" shows "sum ((^) x) {m..n} = (if n < m then 0::'a else if x = (1::'a) then of_nat (n + 1 - m) else (x ^ m - x ^ Suc n) / ((1::'a) - x))"
alw_shift: fixes \<phi> :: "'c stream \<Rightarrow> bool"   and xl :: "'c list"   and xs :: "'c stream" assumes "alw \<phi> (xl @- xs)" shows "alw \<phi> xs"
N_sums: fixes q :: "real"   and n :: "int" assumes "fbnorm q" shows "(\<lambda>k. if k = 0 \<or> int k dvd n then 0 else 1 / q ^ k) sums fbnorm.N q n"
r_child_lower_bound: fixes l :: "int"   and p :: "int" assumes "l \<le> p" shows "p < r_child l p"
safe_distance_2r_def: shows "safe_distance_2r = v\<^sub>e * \<delta> - v\<^sub>e\<^sup>2 / 2 / a\<^sub>e + v\<^sub>o\<^sup>2 / 2 / a\<^sub>o"
openI: fixes S :: "'a set" assumes "\<And>x. x \<in> S \<Longrightarrow> \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S" shows "open S"
assms_2: shows "dim_col A\<^sup>T = dim_vec b"
assms_2: shows "f \<in>\<^sub>f\<^sub>s F"
norm_set_diag: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" assumes "canonical M n"   and "[M]\<^bsub>v,n\<^esub> \<noteq> {}" obtains M' :: "nat \<Rightarrow> nat \<Rightarrow> real DBMEntry" where "[M]\<^bsub>v,n\<^esub> = [M']\<^bsub>v,n\<^esub>" and "[DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub> = [DBM_Normalization.norm M' (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub>" and "\<forall>i\<le>n. M' i i = \<one>" and "canonical M' n"
unique_row_col_1: fixes nr1 :: "nat"   and nc1 :: "nat"   and M :: "'a list list"   and nr2 :: "nat"   and nc2 :: "nat" assumes "mat nr1 nc1 M"   and "mat nr2 nc2 M"   and "M \<noteq> []" shows "nr1 = nr2"
in_lang_INTERSECT: fixes n :: "nat"   and w :: "'a list"   and xs :: "'b rexp list" assumes "wf_word n w" shows "(w \<in> lang n (INTERSECT xs)) = (\<forall>r\<in>set xs. w \<in> lang n r)"
trancl_into_trancl: fixes r :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'e"   and b :: "'e"   and c :: "'e" assumes "r\<^sup>+\<^sup>+ a b"   and "r b c" shows "r\<^sup>+\<^sup>+ a c"
le_boolI: fixes P :: "bool"   and Q :: "bool" assumes "P \<Longrightarrow> Q" shows "P \<le> Q"
swap_StateMerge: fixes a :: "'d \<Longrightarrow> 'e"   and b :: "'f \<Longrightarrow> 'e" assumes "a \<bowtie> b" shows "swap\<^sub>m ;; M[a|b]\<^sub>\<sigma> = M[b|a]\<^sub>\<sigma>"
aHeap_thunks_rec: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and x :: "Vars.var"   and e :: "exp"   and a :: "Arity" assumes "\<not> nonrec \<Gamma>"   and "x \<in> thunks \<Gamma>"   and "x \<in> edom (Aheap \<Gamma> e\<cdot>a)" shows "(Aheap \<Gamma> e\<cdot>a) x = up\<cdot>0"
rev_butlast_is_tl_rev: fixes l :: "'c list" shows "rev (butlast l) = tl (rev l)"
arr_def: fixes f :: "'a1 \<times> 'a2" shows "A1xA2.arr f \<equiv> A1xA2.domains f \<noteq> {} \<or> A1xA2.codomains f \<noteq> {}"
hoare_patial_seq_assoc_sem: fixes dims :: "nat list"   and A :: "complex mat"   and S1 :: "com"   and S2 :: "com"   and S3 :: "com"   and B :: "complex mat" shows "state_sig.hoare_partial_correct dims A (S1;; S2;; S3) B = state_sig.hoare_partial_correct dims A (S1;; (S2;; S3)) B"
interval_real_split_1: fixes a :: "real"   and b :: "real"   and c :: "real" shows "{a..b} \<inter> {x. x \<le> c} = {a..min b c}"
simps_2: fixes e :: "'b"   and a :: "'c"   and r :: "nat"   and ts :: "('b, 'c) SkewBinomialTree list" shows "tree_invar (Node e a (Suc r) ts) = (\<exists>e1 a1 ts1 e2 a2 ts2 e' a'. tree_invar (Node e1 a1 r ts1) \<and> tree_invar (Node e2 a2 r ts2) \<and> (Node e a (Suc r) ts = link (Node e1 a1 r ts1) (Node e2 a2 r ts2) \<or> Node e a (Suc r) ts = skewlink e' a' (Node e1 a1 r ts1) (Node e2 a2 r ts2)))"
assets_def: fixes Mkt :: "'d" shows "assets Mkt = UNIV"
lt_lower_less: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and v :: "'a" assumes "punit.lower p v \<noteq> 0" shows "punit.lt (punit.lower p v) \<prec> v"
inj_imp_surj_inv: fixes f :: "'a \<Rightarrow> 'b" assumes "inj f" shows "surj (inv f)"
subprob_space_distr: fixes f :: "val \<Rightarrow> 'a"   and M' :: "'a measure" assumes "f \<in> M \<rightarrow>\<^sub>M M'"   and "space M' \<noteq> {}" shows "subprob_space (distr M M' f)"
sum_diff1: fixes A :: "'d set"   and f :: "'d \<Rightarrow> 'e"   and a :: "'d" assumes "finite A" shows "sum f (A - {a}) = (if a \<in> A then sum f A - f a else sum f A)"
elims: fixes x :: "('a \<times> 'b) \<times> 'a \<times> 'c"   and y :: "'a" assumes "Pre_Port_Graph.edge_end x = y" obtains v1 :: "'a" and p1 :: "'b" and v2 :: "'a" and p2 :: "'c" where "x = ((v1, p1), v2, p2)" and "y = v2"
finite_Collect_disjI: fixes P :: "'e \<Rightarrow> bool"   and Q :: "'e \<Rightarrow> bool" shows "finite {x. P x \<or> Q x} = (finite {x. P x} \<and> finite {x. Q x})"
separation_closures: fixes S :: "'a set"   and T :: "'a set" assumes "S \<inter> closure T = {}"   and "T \<inter> closure S = {}" obtains U :: "'a set" and V :: "'a set" where "U \<inter> V = {}" and "open U" and "open V" and "S \<subseteq> U" and "T \<subseteq> V"
\<tau>inf_step_table_into_\<tau>inf_step: fixes trsys :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and \<tau>move :: "'g \<Rightarrow> 'h \<Rightarrow> 'g \<Rightarrow> bool"   and s :: "'g"   and sstls :: "('g \<times> 'g \<times> 'h \<times> 'g) llist" assumes "\<tau>trsys.\<tau>inf_step_table trsys \<tau>move s sstls" shows "\<tau>trsys.\<tau>inf_step trsys \<tau>move s (lmap (fst \<circ> snd \<circ> snd) sstls)"
acc_bacc_eq: fixes A :: "('a \<times> 'a) set" assumes "finite A" shows "Wellfounded.acc A = bacc A (card UNIV)"
individually_rational_on_CD_on: fixes ds :: "'d set"   and X :: "'x set" assumes "individually_rational_on ds X" shows "CD_on ds X = X"
monofun_def: fixes R :: "('j, 'k) Ring_scheme"   and M :: "('h, 'j, 'l) Module_scheme"   and N :: "('i, 'j, 'm) Module_scheme"   and f :: "'h \<Rightarrow> 'i"   and H :: "'h set"   and h :: "'h" shows "monofun R M N f H h = (\<lambda>x\<in>fgs R M {h}. THE y. \<exists>r\<in>carrier R. x = r \<cdot>\<^sub>s\<^bsub>M\<^esub> h \<and> y = r \<cdot>\<^sub>s\<^bsub>N\<^esub> f h)"
trm_rep_simp_def: fixes t :: "'a trm"   and S :: "'a eclause set" shows "trm_rep t S = (if subterm_reduction_applicable S t then subterm_reduction S t else get_min t (set_of_candidate_values S t))"
WF_Box: fixes A :: "state \<times> state \<Rightarrow> bool"   and v :: "state \<Rightarrow> 'a" shows "\<turnstile> (\<box>WF(A)_v) = WF(A)_v"
member_le_L2_set: fixes A :: "'a set"   and i :: "'a"   and f :: "'a \<Rightarrow> real" assumes "finite A"   and "i \<in> A" shows "f i \<le> L2_set f A"
integrable_on_superset: fixes f :: "'b \<Rightarrow> 'c"   and S :: "'b set"   and t :: "'b set" assumes "f integrable_on S"   and "\<And>x. x \<notin> S \<Longrightarrow> f x = (0::'c)"   and "S \<subseteq> t" shows "f integrable_on t"
fst_def: fixes p :: "('c, 'd) Tagged_Prod_Sum.prod" shows "Tagged_Prod_Sum.fst p = (case p of prod.Prod x xa a xb \<Rightarrow> a)"
SpanD_lincomb: fixes smult :: "'b \<Rightarrow> 'a \<Rightarrow> 'a"   and ms :: "'a list" shows "R_scalar_mult.RSpan UNIV smult ms = {scalar_mult.lincomb smult rs ms |rs. set rs \<subseteq> UNIV \<and> length rs = length ms}"
opt_mru_vote_def: fixes lvs :: "process \<Rightarrow> (nat \<times> val) option"   and Q :: "process set" shows "opt_mru_vote lvs Q = option_Max_by fst (ran (lvs |` Q))"
rank_less_row_i_imp_i_is_zero: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "'c" assumes "rank A \<le> mod_type_class.to_nat i" shows "Gauss_Jordan A $ i = 0"
cofactor_def: fixes A :: "'b mat"   and i :: "nat"   and j :: "nat" shows "Determinant.cofactor A i j = (- (1::'b)) ^ (i + j) * Determinant.det (mat_delete A i j)"
asBinp_equal_iff_alphaBinp: fixes qbinp :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs option"   and qbinp' :: "'d \<Rightarrow> ('e, 'f, 'varSort, 'var, 'g) qAbs option" assumes "qGoodBinp qbinp" shows "(asBinp qbinp = asBinp qbinp') = ((qbinp, qbinp') \<in> alphaBinp)"
tc_mult_scalar: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.tc (local.punit.mult_scalar p q) = punit.tc p * punit.tc q"
mono_rtranclp: fixes x :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and a :: "'d"   and b :: "'d" assumes "\<And>a b. x a b \<longrightarrow> y a b" shows "x\<^sup>*\<^sup>* a b \<longrightarrow> y\<^sup>*\<^sup>* a b"
inj_op_nth: fixes X :: "(('a, 'b) vec, 'b) vec" assumes "is_basis (set_of_vector X)" shows "inj (($) X)"
zimage_cong: fixes M :: "'a zet"   and N :: "'a zet"   and f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b" assumes "M = N"   and "\<And>x. zin x N \<Longrightarrow> f x = g x" shows "zimage f M = zimage g N"
cong_2: fixes P :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool"   and P' :: "(nat \<Rightarrow> 'b) \<Rightarrow> bool" assumes "\<And>\<sigma>. P \<sigma> = P' \<sigma>" shows "\<box>P = \<box>P'"
add_mono_thms_linordered_semiring_1: fixes i :: "'d"   and j :: "'d"   and k :: "'d"   and l :: "'d" assumes "i \<le> j \<and> k \<le> l" shows "i + k \<le> j + l"
assms_2: fixes x :: "'a vec" assumes "x \<in> set ls" shows "dim_vec x = nr"
integrable_spike_set_eq: fixes S :: "'b set"   and T :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "negligible (S - T \<union> (T - S))" shows "(f integrable_on S) = (f integrable_on T)"
upright_def: fixes a :: "'b zmultiset" shows "upright a = (\<forall>t. 0 < zcount a t \<longrightarrow> supported a t)"
am_add_op_def: fixes x :: "'a"   and y :: "'a" shows "x \<oplus> y \<equiv> ad (ad x \<cdot> ad y)"
push_bit_nonnegative_int_iff: fixes n :: "nat"   and k :: "int" shows "(0 \<le> push_bit n k) = (0 \<le> k)"
chain1_eq_chain2: shows "chain_subdiv_chain one_chain1 subdiv"
mono: fixes x :: "'a llist" shows "lfp.mono_body (\<lambda>f. F f x)"
finite_Union: fixes X :: "'a set set"   and M :: "'a measure" assumes "finite X"   and "X \<subseteq> fmeasurable M" shows "\<Union> X \<in> fmeasurable M"
inj_imp_surj_inv: fixes f :: "'b \<Rightarrow> 'c" assumes "inj f" shows "surj (inv f)"
make_mat_def: fixes Smith_1x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat"   and Smith_2x2 :: "'a mat \<Rightarrow> 'a mat \<times> 'a mat \<times> 'a mat"   and div_op :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and n :: "nat"   and k :: "nat"   and B :: "'a mat" assumes "Smith_Impl Smith_1x2 Smith_2x2 div_op" shows "Smith_Impl.make_mat n k B = Matrix.mat n n (\<lambda>(i, j). if i = 0 \<and> j = 0 then B $$ (0, 0) else if i = 0 \<and> j = k then B $$ (0, 1) else if i = k \<and> j = 0 then B $$ (1, 0) else if i = k \<and> j = k then B $$ (1, 1) else if i = j then 1::'a else (0::'a))"
divide_by_norm_def: fixes A :: "(('c, 'd) vec, 'e) vec" shows "divide_by_norm A = (\<chi>a b. Projections.normalize (column b A) $ a)"
simps_27: fixes f1 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> k \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and f3 :: "(nat \<Rightarrow> nat option) \<Rightarrow> nat \<Rightarrow> 'a"   and f4 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f5 :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> 'a"   and x21 :: "nat"   and x22 :: "nat"   and x23 :: "nat"   and x24 :: "nat"   and x25 :: "nat" shows "(case Rrep x21 x22 x23 x24 x25 of Rreq x xa xb xc xd xe xf xg \<Rightarrow> f1 x xa xb xc xd xe xf xg | Rrep x xa xb xc xd \<Rightarrow> f2 x xa xb xc xd | Rerr x xa \<Rightarrow> f3 x xa | Newpkt x xa \<Rightarrow> f4 x xa | Pkt x xa xb \<Rightarrow> f5 x xa xb) = f2 x21 x22 x23 x24 x25"
VLambda_vconst_on: fixes A :: "V"   and c :: "V" shows "(\<lambda>a\<in>\<^sub>\<circ>A. c) = vconst_on A c"
ord_strict_higher: fixes p :: "'a \<Rightarrow>\<^sub>0 'c"   and q :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.ord_strict_p p q = (\<exists>v. lookup p v = (0::'c) \<and> lookup q v \<noteq> (0::'c) \<and> punit.higher p v = punit.higher q v)"
abs_eq: fixes x :: "int extended \<times> int extended" shows "Abs_Int2_ivl.above (abs_ivl x) = abs_ivl (above_rep x)"
fps_power_Suc_eqD': fixes f :: "'a fps"   and m :: "nat"   and g :: "'a fps" assumes "f ^ Suc m = g ^ Suc m"   and "fps_nth f (subdegree f) = fps_nth g (subdegree g)" shows "f = g"
wf_digraph_del_arc: fixes a :: "'b" shows "wf_digraph (H.del_arc a)"
nv2T: shows "infTp (tpOf T) \<or> (\<forall>x\<in>nv2T T. tpOfV x \<noteq> tpOf T)"
1: shows "[\<^bold>\<box>\<phi> \<alpha> \<^bold>& (\<^bold>\<forall>\<beta>. \<^bold>\<box>\<phi> \<beta> \<^bold>\<rightarrow> \<beta> \<^bold>= \<alpha>) in v]"
switch_leq: fixes u' :: "view"   and u :: "view"   and v :: "view"   and c :: "cars" shows "u' \<le> u \<and> v = c > u \<longrightarrow> (\<exists>v'. v' = c > u' \<and> v' \<le> v)"
simps_6: fixes t :: "'c local.term" shows "Cod \<^bold>\<l>\<^sup>-\<^sup>1\<^bold>[t\<^bold>] = \<^bold>\<I> \<^bold>\<otimes> Cod t"
empty_set_less_aux_finite_iff: fixes A :: "'a set" assumes "finite A" shows "({} \<sqsubset>' A) = (A \<noteq> {})"
length_ltl_plays_from_strategy: fixes \<sigma> :: "'c list \<Rightarrow> 'c"   and p :: "'c list" shows "llength (ltl (strategy_progression \<sigma> p)) = \<infinity>"
assms_4: shows "\<guillemotleft>\<epsilon> : f \<star>\<^sub>C g \<Rightarrow>\<^sub>C src\<^sub>C g\<guillemotright>"
inner_Basis_INF_left: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and dist :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and norm :: "'c \<Rightarrow> real"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and uminus :: "'c \<Rightarrow> 'c"   and scaleR :: "real \<Rightarrow> 'c \<Rightarrow> 'c"   and sgn :: "'c \<Rightarrow> 'c"   and uniformity :: "('c \<times> 'c) filter"   and "open" :: "'c set \<Rightarrow> bool"   and inner :: "'c \<Rightarrow> 'c \<Rightarrow> real"   and Basis :: "'c set"   and abs :: "'c \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and i :: "'c"   and f :: "'d \<Rightarrow> 'c"   and X :: "'d set" assumes "class.ordered_euclidean_space Inf Sup minus dist norm plus zero uminus scaleR sgn uniformity open inner Basis abs inf sup less_eq less"   and "i \<in> Basis" shows "inner (Inf (f ` X)) i = (INF x\<in>X. inner (f x) i)"
taylor_term_def: fixes R :: "('c, 'd) ring_scheme"   and c :: "'c"   and p :: "nat \<Rightarrow> 'c"   and i :: "nat" assumes "UP_cring R" shows "UP_cring.taylor_term R c p i = taylor_expansion R c p i \<odot>\<^bsub>UP R\<^esub> X_poly_minus R c [^]\<^bsub>UP R\<^esub> i"
partList2_empty: fixes ts :: "'b set list" shows "partList2 {} ts = ts"
is_unit_mult_iff: fixes a :: "'a"   and b :: "'a" shows "is_unit (a * b) = (is_unit a \<and> is_unit b)"
simps: fixes P :: "'a \<Rightarrow> bool"   and xs :: "'a list"   and ko :: "'b" shows "filter_raw P (xs, ko) = (filter P xs, ko)"
assms_3: fixes n :: "nat" shows "a (Suc n) \<le> a n"
components_are_iso: fixes a :: "'c \<times> 'c" assumes "C.VV.ide a" shows "D.iso (\<Phi> a)"
simps_5: fixes I :: "('e, 'f, 'd) interp"   and t1 :: "('e, 'd) trm"   and t2 :: "('e, 'd) trm" shows "dterm_sem I (Times t1 t2) = (\<lambda>v. dterm_sem I t1 v * dterm_sem I t2 v)"
all_info2: shows "set (characterize_consistent_signs_at_roots_copr p qs2) \<subseteq> set signs2"
keys_map_scale: fixes k :: "'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "keys (k \<cdot> t) = (if k = (0::'c) then {} else keys t)"
Nat_case_strict: fixes z :: "'a"   and s :: "nat discr Box \<rightarrow> 'a" shows "Nat_case\<cdot>z\<cdot>s\<cdot>\<bottom> = \<bottom>"
is_aligned_addD1: fixes x :: "'b word"   and y :: "'b word"   and n :: "nat" assumes "is_aligned (x + y) n"   and "is_aligned x n" shows "is_aligned y n"
no_trailing_strip_while: fixes P :: "'b \<Rightarrow> bool"   and xs :: "'b list" shows "no_trailing P (strip_while P xs)"
lossless_fdr_step: fixes n :: "nat"   and v :: "nat"   and c :: "nat" shows "lossless_spmf (Fast_Dice_Roll.fdr_step n v c) = (0 < v)"
A: shows "round_suc_suc index key (take (Suc (Suc n)) xs) n n' u = (v, ms', ws')"
lim_proc_is_lub3a: fixes s :: "'a event list"   and P :: "'a process"   and c :: "'a event" assumes "front_tickFree s" shows "s \<notin> D P \<longrightarrow> (\<forall>t. t \<in> D P \<longrightarrow> \<not> t < s @ [c])"
assms_3: shows "e \<le> v ; - v\<^sup>T"
m1_def: shows "m1 \<equiv> \<lparr>init = m1_init, trans = m1_trans, obs = id\<rparr>"
monofun_add: fixes N :: "('g, 'b, 'h) Module_scheme"   and H :: "'a set"   and f :: "'a \<Rightarrow> 'g"   and h :: "'a"   and x :: "'a"   and y :: "'a" assumes "R module N"   and "free_generator R M H"   and "f \<in> H \<rightarrow> carrier N"   and "h \<in> H"   and "x \<in> fgs R M {h}"   and "y \<in> fgs R M {h}" shows "monofun R M N f H h (x \<plusminus> y) = monofun R M N f H h x \<plusminus>\<^bsub>N\<^esub> monofun R M N f H h y"
path_last_succ: fixes A :: "('d, 'e, 'f, 'g) FSM_scheme"   and ws :: "('d \<times> 'e) list"   and r1s :: "'f list"   and p1 :: "'f" assumes "path A (ws || r1s) p1"   and "length r1s = length ws"   and "0 < length ws" shows "last r1s \<in> succ A (last ws) (target (butlast (ws || r1s)) p1)"
prems_1: shows "outstanding_refs is_volatile_Write\<^sub>s\<^sub>b (x # sb) = {}"
simps_4: fixes c :: "'a"   and r1 :: "'a rexp"   and r2 :: "'a rexp" shows "pderiv c (Plus r1 r2) = pderiv c r1 \<union> pderiv c r2"
consistentDP_def: shows "consistentDP \<equiv> ((=) ===>\<^sub>T (=)) dp"
domain_subset_of_D: shows "{v. \<exists>x\<ge>k1. x \<le> k2 \<and> v = x *\<^sub>R i + c} \<subseteq> D"
simps_1: fixes R :: "('b \<times> 'b) set"   and x :: "'b" shows "insert_wrt R x [] = [x]"
assms_5: shows "prv (imp \<phi> (neg (subst \<chi> t x)))"
preimage_included_then_null_or_conull: fixes A :: "'a set" assumes "A \<in> sets M"   and "T --` A \<subseteq> A" shows "A \<in> null_sets M \<or> space M - A \<in> null_sets M"
frechet_derivative: fixes prod :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and x :: "'f \<Rightarrow> 'c"   and a :: "'f"   and y :: "'f \<Rightarrow> 'd" assumes "bounded_bilinear prod"   and "x differentiable at a"   and "y differentiable at a" shows "frechet_derivative (\<lambda>a. prod (x a) (y a)) (at a) = (\<lambda>h. prod (x a) (frechet_derivative y (at a) h) + prod (frechet_derivative x (at a) h) (y a))"
no_Bv_Type_param_irrelevant_typ_of: fixes t :: "Core.term"   and Ts :: "typ list"   and Ts' :: "typ list" assumes "\<not> exists_subterm (\<lambda>x. case x of Bv x \<Rightarrow> True | _ \<Rightarrow> False) t" shows "typ_of1 Ts t = typ_of1 Ts' t"
right_lang_def: fixes q :: "hf" shows "NR.right_lang q \<equiv> {u. NR.nextl {q} u \<inter> nfa.final (Reverse_nfa M) \<noteq> {}}"
ccpo_Sup_upper: fixes A :: "'b set"   and x :: "'b" assumes "Complete_Partial_Order.chain (\<sqsubseteq>) A"   and "x \<in> A" shows "x \<sqsubseteq> \<Or> A"
g_3: fixes x :: "'a metric_completion" assumes "x \<in> range to_metric_completion" shows "g x = f (inv to_metric_completion x)"
union_ring_carrier: fixes C :: "('b, 'c) ring_scheme set" shows "carrier (union_ring C) = \<Union> (carrier ` C)"
source: fixes S :: "'procS"   and TRel :: "('procT \<times> 'procT) set" shows "SourceTerm S \<lesssim>\<lbrakk>\<cdot>\<rbrakk>RT<TRel> SourceTerm S"
hitting_Q_or_new_last_def: shows "hitting_Q_or_new_last \<equiv> \<lambda>y. y \<noteq> v0 \<and> (y = new_last \<or> (\<exists>Q_hit\<in>Q. y \<in> set Q_hit))"
Infinitesimal_hRe: fixes x :: "complex star" assumes "x \<in> Infinitesimal" shows "hRe x \<in> Infinitesimal"
finsum_infinite: fixes A :: "'f set"   and f :: "'f \<Rightarrow> 'c" assumes "infinite A" shows "finsum M f A = \<zero>\<^bsub>M\<^esub>"
min_for_def: fixes a :: "'a"   and b :: "'a" shows "min_for a b = (if le a b then a else b)"
gr_implies_not_zero: fixes m :: "'c multiset"   and n :: "'c multiset" assumes "m \<subset># n" shows "n \<noteq> {#}"
floor_less_zero: fixes x :: "'c" shows "(\<lfloor>x\<rfloor> < 0) = (x < (0::'c))"
u_simps_3: shows "trg s\<^sub>0 = src s"
CptnModWhile2: fixes P :: "'b com"   and s :: "'b"   and xs :: "('b com option \<times> 'b) list"   and b :: "'b set"   and zs :: "('b com option \<times> 'b) list"   and ys :: "('b com option \<times> 'b) list" assumes "(Some P, s) # xs \<in> cptn_mod"   and "fst (last ((Some P, s) # xs)) = None"   and "s \<in> b"   and "zs = map (lift (While b P)) xs @ ys"   and "(Some (While b P), snd (last ((Some P, s) # xs))) # ys \<in> cptn_mod" shows "(Some (While b P), s) # (Some (Seq P (While b P)), s) # zs \<in> cptn_mod"
leg0_in_hom_2: shows "\<guillemotleft>r\<^sub>0s\<^sub>1.p\<^sub>0 : r\<^sub>0s\<^sub>1.p\<^sub>0 \<Rightarrow> r\<^sub>0s\<^sub>1.p\<^sub>0\<guillemotright>"
dim_vec_conjugate: fixes v :: "'c vec" shows "dim_vec (conjugate v) = dim_vec v"
elims_2: fixes x :: "const"   and xa :: "const" assumes "x \<le> xa" obtains xb :: "const" and y :: "const" where "x = xb" and "xa = y" and "y = Any \<or> xb = y"
hyps_5: shows "\<exists>xs e. mbody(CT,m,D) = xs . e \<and> length xs = length Cs"
rbt_bulkload_def: fixes less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "('b \<times> 'c) list" shows "ord.rbt_bulkload less xs = foldr (\<lambda>(k, v). ord.rbt_insert less k v) xs rbt.Empty"
ss43iii: fixes x :: "'a" shows "is_map x = (\<forall>y. x ; - y = - (x ; y))"
elims: fixes x :: "'c ltl list"   and y :: "'c ltl" assumes "And x = y" obtains "x = []" and "y = true"   | xa :: "'c ltl" and xs :: "'c ltl list" where "x = xa # xs" and "y = foldl LTLAnd xa xs"
dist_triangle2: fixes dist :: "'g \<Rightarrow> 'g \<Rightarrow> real"   and uniformity :: "('g \<times> 'g) filter"   and "open" :: "'g set \<Rightarrow> bool"   and x :: "'g"   and y :: "'g"   and z :: "'g" assumes "class.metric_space dist uniformity open" shows "dist x y \<le> dist x z + dist y z"
rel_carrier_Eps_in_1: fixes X :: "('c set \<times> 'd) set" assumes "X \<in> stfx.carrier_direct_lim" shows "(SOME x. x \<in> X) \<in> X"
domain_eq: shows "Domainp pcr_real_int = (\<lambda>r. fst r \<le> snd r)"
cut_lessI: fixes x :: "'a"   and I :: "'a set"   and t :: "'a" assumes "x \<in> I"   and "x < t" shows "x \<in> I \<down>< t"
matrix_sum_distrib_left: fixes P :: "complex mat"   and d :: "nat"   and n :: "nat"   and f :: "nat \<Rightarrow> complex mat" assumes "P \<in> carrier_mat d d"   and "\<And>k. k < n \<Longrightarrow> f k \<in> carrier_mat d d" shows "matrix_sum d (\<lambda>k. P * f k) n = P * matrix_sum d f n"
pa_def: shows "pa \<equiv> \<Prod>j = 1..n. real_of_int (a j)"
smult_is_Zp_cauchy: fixes s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and a :: "nat \<Rightarrow> int" assumes "is_Zp_cauchy s"   and "a \<in> carrier Zp" shows "is_Zp_cauchy (a \<odot>\<^bsub>Zp\<^bsup>\<omega>\<^esup>\<^esub> s)"
is_lasso_prpl_def: fixes m :: "'a"   and prpl :: "'v list \<times> 'v list" shows "gb_graph.is_lasso_prpl (to_gbg_ext m) prpl \<equiv> graph.is_lasso_prpl_pre (to_gbg_ext m) prpl \<and> (\<forall>A\<in>gbg_F (to_gbg_ext m). set (snd prpl) \<inter> A \<noteq> {})"
semantics_in_vertices_1: fixes A :: "('d, 'e) labeled_graph"   and a :: "'e"   and b :: "'e"   and e :: "'d allegorical_term" assumes "graph A"   and "(a, b) \<in> :A:\<lbrakk>e\<rbrakk>" shows "a \<in> vertices A"
elims_3: fixes x :: "('a \<times> ('b \<times> 'c list) list) list" assumes "\<not> exe_ars_conds' x" obtains arss :: "('a \<times> ('b \<times> 'c list) list) list" where "x = arss" and "\<not> (alist_conds arss \<and> (\<forall>ars\<in>snd ` set arss. alist_conds ars))"
nMT_domMT: fixes p :: "((int \<times> int \<times> protocol) set set, protocol \<times> int) Combinators list"   and r :: "int \<times> (int \<times> int \<times> protocol) \<times> (int \<times> int \<times> protocol) \<times> DummyContent" assumes "\<not> not_MT Cp p"   and "p \<noteq> []" shows "r \<notin> dom (Cp (list2FWpolicy p))"
assms_2: fixes x :: "nat" assumes "x \<in> S" shows "p x \<noteq> x"