hcju/isabelleps huggingface dataset:
sel_1: fixes ws :: "'d list stream" shows "shd (flat ws) = hd (shd ws)"
rank_invar_cons: fixes t :: "('a, 'b) BinomialHeapStruc.BinomialTree"   and bq :: "('a, 'b) BinomialHeapStruc.BinomialTree list" assumes "BinomialHeapStruc.rank_invar (t # bq)" shows "BinomialHeapStruc.rank_invar bq"
hyps: fixes "is" :: "'a set" assumes "\<And>n. n < length ias \<Longrightarrow> interrupt_action_ok (redT_updIs is (take n ias)) (ias ! n)" shows "interrupt_actions_ok is ias"
intro: fixes M :: "'c measure" assumes "finite_measure M"   and "subprob_space_axioms M" shows "subprob_space M"
IH: fixes v :: "'a list"   and e :: "nat list \<Rightarrow> 'a" assumes "length v = prod_list ds"   and "\<And>is. is \<lhd> ds \<Longrightarrow> lookup_base ds v is = e is" shows "tensor_vec_from_lookup ds e = v"
AFix_nonrec_eqvt: fixes \<pi> :: "perm"   and cccExp :: "exp \<Rightarrow> Arity \<rightarrow> (Vars.var \<Rightarrow> Arity\<^sub>\<bottom>) \<times> CoCalls"   and x :: "Vars.var"   and e :: "exp" shows "\<pi> \<bullet> CoCallArityAnalysis.Afix_nonrec cccExp x e = CoCallArityAnalysis.Afix_nonrec (\<pi> \<bullet> cccExp) (\<pi> \<bullet> x) (\<pi> \<bullet> e)"
reachableE: fixes u :: "'c"   and v :: "'c"   and G :: "('c, 'd) pre_digraph" assumes "u \<rightarrow>\<^bsub>G\<^esub> v" obtains e :: "'d" where "e \<in> arcs G" and "tail G e = u" and "head G e = v"
split: fixes P :: "'d \<Rightarrow> bool"   and f1 :: "'d"   and f2 :: "nat \<Rightarrow> 'd"   and nat :: "nat" shows "P (case nat of 0 \<Rightarrow> f1 | Suc x \<Rightarrow> f2 x) = ((nat = 0 \<longrightarrow> P f1) \<and> (\<forall>x2. nat = Suc x2 \<longrightarrow> P (f2 x2)))"
terminates_restrict_to_terminates: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and M :: "'d set"   and c :: "('e, 'd, 'f) com"   and s :: "('e, 'f) xstate" assumes "\<Gamma>|\<^bsub>M\<^esub>\<turnstile>c \<down> s"   and "\<Gamma>|\<^bsub>M\<^esub>\<turnstile> \<langle>c,s\<rangle> \<Rightarrow>\<notin>{Stuck}" shows "\<Gamma>\<turnstile>c \<down> s"
res: shows "(if i < m \<and> LLL_invariant upw i fs then basis_reduction_main (basis_reduction_step upw i fs) else fs) = fs'"
fun_sum_apply: fixes u :: "'d \<Rightarrow> 'e \<Rightarrow> 'c"   and I :: "'d set"   and x :: "'e" shows "sum u I x = (\<Sum>i\<in>I. u i x)"
add: fixes f :: "'c \<Rightarrow> real"   and g :: "'c \<Rightarrow> real" assumes "real_polynomial_function f"   and "real_polynomial_function g" shows "real_polynomial_function (\<lambda>x. f x + g x)"
path_join_eq: fixes g1 :: "real \<Rightarrow> 'a"   and g2 :: "real \<Rightarrow> 'a" assumes "path g1"   and "path g2" shows "path (g1 +++ g2) = (pathfinish g1 = pathstart g2)"
prikey_unused: fixes s :: "(agent \<times> msg) set" assumes "s\<^sub>0 \<Turnstile> s" shows "\<exists>A. PriKey A \<notin> used s"
Nats_cases: fixes x :: "'a" assumes "x \<in> \<nat>" obtains n :: "nat" where "x = of_nat n"
simps_3: fixes P :: "'a pred"   and xq :: "'a seq" shows "pred_of_seq (Join P xq) = P \<squnion> pred_of_seq xq"
add_minus: fixes x :: "'a" assumes "x \<in> V" shows "x + - x = (0::'a)"
mem_Ch_CH: fixes x :: "'a"   and h :: "'c"   and X :: "'a set" assumes "x \<in> BCh h X" shows "x \<in> Bossy.CH X"
mult_pCons_left: fixes a :: "'c"   and p :: "'c poly"   and q :: "'c poly" shows "pCons a p * q = Polynomial.smult a q + pCons (0::'c) (p * q)"
simps_7: fixes \<phi> :: "'a ltln"   and \<xi> :: "'a ltln" shows "suspendable (\<phi> W\<^sub>n \<xi>) = (suspendable \<phi> \<and> suspendable \<xi> \<or> pure_eventual \<phi> \<and> \<xi> = false\<^sub>n)"
sd2_at_most_sd4: assumes "a\<^sub>e < a\<^sub>o" shows "safe_distance_2r \<le> safe_distance_4r"
simps: fixes f :: "int vec list \<times> int vec list"   and mu :: "int iarray iarray"   and d :: "int iarray"   and i :: "nat" shows "d_state (f, mu, d) i = d !! i"
INV: fixes l1i :: "'a list"   and xi :: "'a"   and l2i :: "'a list"   and si :: "'c" assumes "li = l1i @ xi # l2i"   and "I l1i (xi # l2i) si" shows "fi xi si \<le>\<^sub>n SPEC (I (l1i @ [xi]) l2i)"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'b"   and color :: "color" shows "P (case color of color.R \<Rightarrow> f1 | color.B \<Rightarrow> f2) = ((color = color.R \<longrightarrow> P f1) \<and> (color = color.B \<longrightarrow> P f2))"
remdups_mset_def: fixes S :: "'c multiset" shows "remdups_mset S = mset_set (set_mset S)"
Compl_insert: fixes x :: "'a"   and A :: "'a set" shows "- insert x A = - A - {x}"
irreducible_m_Mp: fixes m :: "int"   and f :: "int poly" shows "poly_mod.irreducible_m m (poly_mod.Mp m f) = poly_mod.irreducible_m m f"
66: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "x \<sqinter> - - y \<squnion> x \<sqinter> - (- y \<sqinter> z) = x \<sqinter> - (- y \<sqinter> z)"
simps_3: fixes n :: "nat"   and f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and y :: "'a"   and x :: "'a" assumes "n \<noteq> 0"   and "even n" shows "efficient_power f y x n = efficient_power f y (f x x) (n div 2)"
assms_3: fixes \<pi>l :: "('state \<times> 'aprop set) stream list" assumes "ev \<phi> (map stl \<pi>l)"   and "\<chi> (map stl \<pi>l)" shows "\<chi> \<pi>l"
remove1_filter_not: fixes P :: "'g \<Rightarrow> bool"   and x :: "'g"   and xs :: "'g list" assumes "\<not> P x" shows "remove1 x (filter P xs) = filter P xs"
IH: fixes u :: "nat" assumes "isPath u es v" shows "(u, v) \<in> E\<^sup>*"
Crypt_synth: fixes K :: "key"   and X :: "msg"   and H :: "msg set" assumes "Crypt K X \<in> synth H" obtains "Crypt K X \<in> H"   | "X \<in> synth H" and "Key K \<in> H"
assms_1: shows "j cd\<^bsup>\<pi>\<^esup>\<rightarrow> i"
T_size_tree: fixes t :: "'b tree" shows "T_size_tree t = 2 * size t + 1"
IH: fixes q :: "'alt pmf" assumes "sd_chain (Suc k) = Some q"   and "sd_chain n = Some s" shows "q \<prec>[Pareto (SD \<circ> R)] s"
vsv_vrestriction: fixes Aa :: "V"   and A :: "V" shows "vsv ((r \<restriction>\<^sup>l\<^sub>\<circ> Aa) \<restriction>\<^sub>\<circ> A)"
U_2: shows "[DBM_Normalization.norm M (\<lambda>x. real ((k \<circ> v') x)) n]\<^bsub>v,n\<^esub> = \<Union> U"
fps_inverse_deriv: fixes a :: "'a fps" assumes "fps_nth a 0 \<noteq> (0::'a)" shows "fps_deriv (inverse a) = - fps_deriv a * (inverse a)\<^sup>2"
1: shows "\<forall>v\<in>fv s. snd v \<le> max_var_set (fv s)"
norm_sgn: fixes x :: "'a" shows "norm (sgn x) = (if x = (0::'a) then 0 else 1)"
vars: shows "free_vars e \<subseteq> shift_var_set V"
dnso5: fixes x :: "'a"   and y :: "'a" shows "d x \<cdot> d y = d y \<cdot> d x"
inv_del: fixes t :: "('b \<times> color) tree"   and x :: "'b" assumes "invh t"   and "invc t" shows "invh (RBT_Set2.del x t) \<and> (color t = Red \<longrightarrow> bheight (RBT_Set2.del x t) = bheight t \<and> invc (RBT_Set2.del x t)) \<and> (color t = Black \<longrightarrow> bheight (RBT_Set2.del x t) = bheight t - 1 \<and> invc2 (RBT_Set2.del x t))"
simps_11: fixes e1 :: "'b aexp"   and e2 :: "'b aexp"   and b2 :: "bool" shows "GExp_Lexorder.less_gexp_aux (Gt e1 e2) (Bc b2) = False"
intro: fixes type_wf :: "(_) heap \<Rightarrow> bool"   and set_tag_name :: "(_) element_ptr \<Rightarrow> char list \<Rightarrow> ((_) heap, exception, unit) prog"   and set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_disconnected_nodes :: "(_) document_ptr \<Rightarrow> ((_) heap, exception, (_) node_ptr list) prog"   and get_disconnected_nodes_locs :: "(_) document_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" assumes "l_set_tag_name\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs"   and "l_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf get_disconnected_nodes get_disconnected_nodes_locs" shows "l_set_tag_name_get_disconnected_nodes\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M type_wf set_tag_name set_tag_name_locs get_disconnected_nodes get_disconnected_nodes_locs"
inf_compl_bot: fixes x :: "'b \<Rightarrow> 'c \<Rightarrow> bool" shows "x \<sqinter> x\<^sup>c = \<delta>"
borel_measurable_Max: fixes I :: "'c set"   and f :: "'c \<Rightarrow> 'd \<Rightarrow> 'e"   and M :: "'d measure" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> f i \<in> borel_measurable M" shows "(\<lambda>x. MAX i\<in>I. f i x) \<in> borel_measurable M"
span_is_subset: fixes S :: "'c set"   and N :: "'c set" assumes "S \<subseteq> N"   and "LinearCombinations.submodule R N M" shows "local.span S \<subseteq> N"
IO: fixes out :: "'i"   and c :: "('h, 'i, 'j) rpv"   and gpv :: "('h, 'i, 'j) gpv"   and input :: "'j"   and \<Gamma> :: "('i, 'j) \<I>"   and x :: "'h" assumes "IO out c \<in> set_spmf (the_gpv gpv)"   and "input \<in> responses_\<I> \<Gamma> out"   and "x \<in> results_gpv \<Gamma> (c input)" shows "x \<in> results_gpv \<Gamma> gpv"
strict_boundedE: fixes b :: "'c"   and c :: "'c"   and a :: "'c" assumes "max b c < a" obtains "b < a" and "c < a"
smap_shift: fixes f :: "'c \<Rightarrow> 'b"   and xs :: "'c list"   and s :: "'c stream" shows "smap f (xs @- s) = map f xs @- smap f s"
context_left: fixes t :: "('a, 'b) tm"   and s :: "('a, 'b) tm"   and u :: "('a, 'b) tm" assumes "t \<rightarrow>\<^sub>e\<^sub>m\<^sub>b s" shows "App t u \<rightarrow>\<^sub>e\<^sub>m\<^sub>b App s u"
dagger_ext: fixes x :: "'a list set" shows "x \<subseteq> p_star x"
mojmir_accept_iff_token_set_accept: shows "\<MM>.accept\<^sub>R' = (\<exists>i<\<MM>.max_rank\<^sub>R. finite \<MM>.fail \<and> finite (\<MM>.merge i) \<and> infinite (\<MM>.succeed i))"
rfn_tool2: fixes a :: "nat"   and s :: "nat" assumes "a < s" shows "a \<le> s - Suc 0"
Least_Suc: fixes P :: "nat \<Rightarrow> bool"   and n :: "nat" assumes "P n"   and "\<not> P 0" shows "(LEAST n. P n) = Suc (LEAST m. P (Suc m))"
span_add_eq: fixes x :: "'c"   and S :: "'c set"   and y :: "'c" assumes "x \<in> cspan S" shows "(x + y \<in> cspan S) = (y \<in> cspan S)"
zcf_zero: shows "zcf \<zero>\<^bsub>P\<^esub> = \<zero>"
size_def: fixes bound :: "'a bound" shows "bounded_index.size bound \<equiv> index_class.size bound"
rewrite_X_def: fixes \<phi> :: "'a ltln" shows "rewrite_X \<phi> = to_ltln (rewrite_X_enat \<phi>)"
dom_char: fixes Arr :: "'a set"   and f :: "(bool, 'a list) concrete_category.arr" shows "partial_magma.dom (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) f = (if partial_magma.arr (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)) f then if f = quiver.One then quiver.One else quiver.Zero else partial_magma.null (concrete_category.COMP {False, True} (\<lambda>x y. Collect (graph.path {False, True} Arr (\<lambda>_. False) (\<lambda>_. True) x y)) (\<lambda>_ _ _ g f. f @ g)))"
lens_override_def: fixes X :: "'c \<Longrightarrow> 'b"   and S\<^sub>1 :: "'b"   and S\<^sub>2 :: "'b" shows "S\<^sub>1 \<triangleleft>\<^bsub>X\<^esub> S\<^sub>2 = put\<^bsub>X\<^esub> S\<^sub>1 (get\<^bsub>X\<^esub> S\<^sub>2)"
M: shows "timpl_closure_set M TI \<turnstile>\<^sub>c T ! (R ! i)"
norm_add_Pythagorean: fixes a :: "'a"   and b :: "'a" assumes "orthogonal a b" shows "(norm (a + b))\<^sup>2 = (norm a)\<^sup>2 + (norm b)\<^sup>2"
in_hhomI: fixes \<mu> :: "'a"   and a :: "'a"   and b :: "'a" assumes "S.arr \<mu>"   and "S.src \<mu> = a"   and "S.trg \<mu> = b" shows "S.in_hhom \<mu> a b"
inv\<^sub>2_def: fixes T :: "nat \<Rightarrow> nat"   and A :: "nat \<Rightarrow> nat set"   and j :: "nat" shows "inv\<^sub>2 T A j = (lb T A j \<and> j \<le> n \<and> (\<forall>T' A'. lb T' A' j \<longrightarrow> real (makespan T) \<le> 3 / 2 * real (makespan T')) \<and> (\<forall>x>j. T x = 0) \<and> (j \<le> m \<longrightarrow> makespan T = Max\<^sub>0 (t ` {1..j})))"
surjective: fixes r :: "('b, 'c) state_scheme" shows "r = \<lparr>m = m r, l = l r, carrots = carrots r, hare = hare r, tortoise = tortoise r, \<dots> = more r\<rparr>"
U_idem_s_prod: shows "U \<cdot> U = U"
real_abs_infnorm: fixes x :: "'a" shows "\<bar>infnorm x\<bar> = infnorm x"
u_def: shows "u = (\<lambda>k. rep_metric_completion (X k) (N k))"
retract_of_contractible: fixes T :: "'b set"   and S :: "'b set" assumes "contractible T"   and "S retract_of T" shows "contractible S"
UNF_onE: fixes r :: "('b \<times> 'b) set"   and A :: "'b set"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "UNF_on r A"   and "a \<in> A" obtains "b = c"   | "(a, b) \<notin> r\<^sup>!"   | "(a, c) \<notin> r\<^sup>!"
assms_1: shows "H \<turnstile> SeqAppendP f1 (SUCC k1) f2 (SUCC k2) g"
sgnxl: shows "(f has_sgnx l) (at_left a)"
add_diff_assoc: fixes a :: "'b"   and b :: "'b"   and c :: "'b" assumes "a \<le> b" shows "c + (b - a) = c + b - a"
prems_2: shows "y \<notin> f ` set (a # xs)"
f2: shows "(r' |\<^bsub>rel\<^esub> s') \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (r |\<^bsub>rel\<^esub> s) = (r' \<otimes> r |\<^bsub>rel\<^esub> s' \<otimes> s)"
r_one: fixes x :: "'a set" assumes "x \<in> carrier (R Quot I)" shows "x \<otimes>\<^bsub>R Quot I\<^esub> \<one>\<^bsub>R Quot I\<^esub> = x"
smult_mat_def: fixes a :: "'b"   and A :: "'b mat" shows "a \<cdot>\<^sub>m A \<equiv> map_mat ((*) a) A"
Orth_def: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and U :: "'p"   and V :: "'p" shows "A B C Orth U V \<equiv> \<exists>X. X OrthAt A B C U V"
max_x_impl: fixes a :: "nat list"   and b :: "nat list"   and y :: "nat list"   and i :: "nat" shows "max_x_impl a b y i = hlde_ops.max_x a b y i"
c3: shows "\<forall>\<psi>\<in>set ns. w \<Turnstile>\<^sub>n F\<^sub>n (G\<^sub>n \<psi>[set ms]\<^sub>\<Pi>\<^sub>1)"
weak_ranking_eqclass2: fixes A :: "'a set"   and x :: "'a"   and y :: "'a" assumes "A \<in> set (weak_ranking le)"   and "x \<in> A"   and "le x y"   and "le y x" shows "y \<in> A"
not_eq_complement: fixes 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 divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bit :: "'a \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and drop_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and take_bit :: "nat \<Rightarrow> 'a \<Rightarrow> 'a"   and "and" :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and or :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and xor :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and mask :: "nat \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and not :: "'a \<Rightarrow> 'a"   and a :: "'a" assumes "class.ring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask uminus not" shows "not a = minus (uminus a) one"
pure_eq_iff: fixes h :: "'l"   and f :: "('l, 'n, 'm) prog" shows "(\<forall>h' x. h \<turnstile> f \<rightarrow>\<^sub>r x \<longrightarrow> h \<turnstile> f \<rightarrow>\<^sub>h h' \<longrightarrow> h = h') = pure f h"
Diagonal_to_Smith_row_i_def: fixes A :: "(('a, 'b) vec, 'c) vec"   and i :: "nat"   and bezout :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<times> 'a \<times> 'a \<times> 'a \<times> 'a" shows "Diagonal_to_Smith_row_i A i bezout = diagonal_to_Smith_i [i + 1..<min (nrows A) (ncols A)] A i bezout"
group_axioms_def: fixes G :: "('a, 'b) monoid_scheme" shows "Group.group_axioms G \<equiv> carrier G \<subseteq> Units G"
assms_1: shows "ModalSequents.mset c = \<LM> A  \<RM>"
simps: fixes s :: "'c state"   and s' :: "'a state"   and res :: "'b" shows "fix_clock s (s', res) = (update_clock (\<lambda>_. if clock s' \<le> clock s then clock s' else clock s) s', res)"
fbox_def: fixes x :: "'a"   and y :: "'a" shows "antirange_semiring.bbox (\<cdot>) ar x y = ar (ar y \<cdot> x)"
adds_pp_splus: fixes pair_of_term :: "'a \<Rightarrow> 'b \<times> 'c"   and term_of_pair :: "'b \<times> 'c \<Rightarrow> 'a"   and t :: "'b"   and v :: "'a"   and s :: "'b" assumes "term_powerprod pair_of_term term_of_pair"   and "term_powerprod.adds_pp pair_of_term t v" shows "term_powerprod.adds_pp pair_of_term t (term_powerprod.splus pair_of_term term_of_pair s v)"
strict_implies_order: fixes a :: "'a list"   and b :: "'a list" assumes "strict_border a b" shows "border a b"
region_def: shows "Regions.region \<equiv> \<lambda>X I r. {x. Regions.regionp (\<lambda>x. x \<in> X) I (\<lambda>x xa. (x, xa) \<in> r) x}"
product_part_in_first_factor: fixes x :: "i"   and y :: "i"   and z :: "i" assumes "O x y"   and "P z (x \<otimes> y)" shows "P z x"
clearjunk_keys_set: fixes al :: "('c \<times> 'd) list" shows "set (map fst (clearjunk al)) = set (map fst al)"
hyps_4: shows "Q \<sim>\<lbrakk>\<cdot>\<rbrakk><SRel,TRel> R"
Class_eq: fixes S :: "'c set"   and E :: "('c \<times> 'c) set"   and a :: "'c"   and b :: "'c" assumes "equivalence S E"   and "(a, b) \<in> E" shows "equivalence.Class S E a = equivalence.Class S E b"
simps_2: fixes X :: "'c event set"   and y :: "'c event"   and t :: "'c event list" shows "setinterleaving ([], X, y # t) = (if y \<in> X then {} else {y # u |u. u setinterleaves (([], t), X)})"
times_in_PolysD: fixes p :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and X :: "'a set" assumes "p * q \<in> P[X]"   and "p \<in> P[X]"   and "p \<noteq> 0" shows "q \<in> P[X]"
assms_3: fixes as :: "'a stream"   and tr :: "'a treeFI" assumes "phi as tr" shows "\<exists>tr'\<in>set (sub tr). phi (stl as) tr'"
totient_power: fixes m :: "nat"   and n :: "nat" assumes "0 < m" shows "totient (n ^ m) = n ^ (m - 1) * totient n"
meet_lower2: fixes x :: "'c"   and y :: "'c" shows "x \<sqinter> y \<sqsubseteq> y"
fin_join_ti_hint1: fixes s :: "'a list list"   and x :: "nat"   and i :: "nat" assumes "fin_join_ti s x (Suc i) = []" shows "fin_join_ti s x i = []"
content_decompose: fixes p :: "'b poly" obtains p' :: "'b poly" where "p = smult (content p) p'" and "content p' = (1::'b)"
simps_2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and as :: "'a list"   and b :: "'b"   and bs :: "'b list" shows "list_all_zip P (a # as) (b # bs) = (P a b \<and> list_all_zip P as bs)"
antisym: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a"   and b :: "'a" assumes "ordering less_eq less"   and "less_eq a b"   and "less_eq b a" shows "a = b"
stdout_world2: shows "literal.explode STR ''Hello World! What is your name?'' = ''Hello World! What is your name?''"
PiE_arb: fixes f :: "'c \<Rightarrow> 'd"   and S :: "'c set"   and T :: "'c \<Rightarrow> 'd set"   and x :: "'c" assumes "f \<in> Pi\<^sub>E S T"   and "x \<notin> S" shows "f x = undefined"
proves_eta_norm: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and \<Gamma> :: "Core.term set"   and t :: "Core.term"   and u :: "Core.term" assumes "wf_theory \<Theta>"   and "finite \<Gamma>"   and "term_ok \<Theta> t"   and "eta_norm t = u"   and "\<forall>A\<in>\<Gamma>. term_ok \<Theta> A"   and "\<forall>A\<in>\<Gamma>. typ_of A = Some propT" shows "\<Theta>,\<Gamma> \<turnstile> mk_eq t u"
sig_trd_spp_aux_simps: fixes bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and v :: "'t"   and p :: "'a \<Rightarrow>\<^sub>0 'b"   and r :: "'a \<Rightarrow>\<^sub>0 'b" shows "sig_trd_spp_aux bs v (p, r) = (if p = 0 then r else sig_trd_spp_aux bs v (sig_trd_spp_body bs v (p, r)))"
MakeCatComp2: fixes C :: "('a, 'b, 'c) Category_scheme"   and f :: "'b"   and g :: "'b" assumes "f \<approx>>\<^bsub>MakeCat C\<^esub> g" shows "f ;;\<^bsub>MakeCat C\<^esub> g = f ;;\<^bsub>C\<^esub> g"
lessalive_trans: fixes s :: "Store"   and t :: "Store"   and w :: "Store" assumes "s \<lless> t"   and "t \<lless> w" shows "s \<lless> w"
f7: shows "\<forall>x f. ((\<exists>xa. (pre xa \<and> (xa, x) \<in> V) \<and> \<not> f xa \<le> (if nofail (f xa) then M xa else FAIL)) \<or> \<not> pre x \<or> REC\<^sub>T body \<noteq> f) \<or> body f x \<le> (if nofail (body f x) then M x else FAIL)"
finite_cspan_closed_csubspace: fixes S :: "'b set" assumes "finite S" shows "closed_csubspace (cspan S)"
simps_3: fixes nm_name :: "char list"   and nm_receiver_violation :: "bool"   and nm_default :: "'a"   and nm_sinvar :: "'b list_graph \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> bool"   and nm_offending_flows :: "'b list_graph \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> ('b \<times> 'b) list list"   and nm_node_props :: "('b, 'a) TopoS_Params \<Rightarrow> 'b \<Rightarrow> 'a"   and nm_eval :: "'b list_graph \<Rightarrow> ('b, 'a) TopoS_Params \<Rightarrow> bool"   and more :: "'c" shows "nm_default \<lparr>nm_name = nm_name, nm_receiver_violation = nm_receiver_violation, nm_default = nm_default, nm_sinvar = nm_sinvar, nm_offending_flows = nm_offending_flows, nm_node_props = nm_node_props, nm_eval = nm_eval, \<dots> = more\<rparr> = nm_default"
simps_28: fixes x5 :: "('b, 'c) fm"   and x3 :: "('b, 'c) fm" shows "(\<^bold>\<diamond> x5) \<noteq> (\<^bold>\<not> x3)"
stock_pf_uvp_expand: fixes pf :: "'a \<Rightarrow> nat \<Rightarrow> bool stream \<Rightarrow> real"   and n :: "nat"   and w :: "bool stream" assumes "stock_portfolio Mkt pf" shows "cls_val_process Mkt pf (Suc n) w = geom_proc (Suc n) w * pf stk (Suc n) w + disc_rfr_proc r (Suc n) w * pf risk_free_asset (Suc n) w"
map_fst_pair_pmf: fixes A :: "'c pmf"   and B :: "'d pmf" shows "map_pmf fst (pair_pmf A B) = A"
gfp_fixpoint: fixes f :: "'d \<Rightarrow> 'd" assumes "mono f" shows "f (\<mu> f) = \<mu> f"
weak_Sup_distl: fixes Y :: "'a set"   and x :: "'a" assumes "Y \<noteq> {}" shows "x \<cdot> \<Squnion> Y = \<Squnion> ((\<cdot>) x ` Y)"
f2: fixes x :: "atom" assumes "supp T - atom ` xs = supp S - atom ` ys"   and "x \<in> atom ` ys"   and "x \<in> supp S" shows "x \<sharp> f xs T"
count_take_less: fixes n :: "nat"   and m :: "nat"   and cs :: "'a list"   and x :: "'a" assumes "n \<le> m" shows "count_list (take n cs) x \<le> count_list (take m cs) x"
a_Lim: shows "Sup (emeasure M ` {Q \<in> sets M. emeasure N Q \<noteq> \<infinity>}) = (SUP i. emeasure M (Q' i))"
set_concat: fixes xs :: "'a list list" shows "set (concat xs) = \<Union> (set ` set xs)"
prv_imp_neg_allI: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla"   and t :: "'trm"   and x :: "'var" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "t \<in> trm"   and "x \<in> var"   and "prv (imp \<phi> (neg (subst \<chi> t x)))" shows "prv (imp \<phi> (neg (all x \<chi>)))"
e_is_trap_def: fixes e :: "e" shows "e_is_trap e = (case e of Trap \<Rightarrow> True | _ \<Rightarrow> False)"
2: shows "(\<Sum>i<n. if i = a then f i else (0::'b)) = (if a = a then f a else (0::'b))"
balanced_net_shallow_model': fixes Z :: "nat"   and M :: "nat"   and N :: "nat" shows "balanced_net (shallow_model' Z M N)"
add_local_ring_at_def: shows "add_local_ring_at \<equiv> local.add_rel"
restrict_indets_def: shows "restrict_indets = poly_subst (\<lambda>x. monomial (1::'d) (restrict_indets_subst x))"
D_props: shows "\<forall>x\<in>{0..1} - s. (\<phi> has_vector_derivative D x) (at x)"
removen_length: fixes n :: "nat"   and xs :: "'b list" shows "length (removen n xs) = (if length xs \<le> n then length xs else length xs - 1)"
Units_bijective: shows "transformation.Sym = {\<alpha> \<in> G \<rightarrow>\<^sub>E G. bij_betw \<alpha> G G}"
cod_runit: fixes a :: "'a" assumes "ide a" shows "cod \<r>[a] = a"
Proc_CFG_CallSkip: fixes p :: "char list"   and es :: "expr list"   and rets :: "char list list" shows "Call p es rets \<turnstile> Label 1 -IEdge \<Up>id\<rightarrow>\<^sub>p Exit"
measure_space_eq: fixes A :: "'c set set"   and \<Omega> :: "'c set"   and \<mu> :: "'c set \<Rightarrow> ennreal"   and \<mu>' :: "'c set \<Rightarrow> ennreal" assumes "A \<subseteq> Pow \<Omega>"   and "\<And>a. a \<in> sigma_sets \<Omega> A \<Longrightarrow> \<mu> a = \<mu>' a" shows "measure_space \<Omega> (sigma_sets \<Omega> A) \<mu> = measure_space \<Omega> (sigma_sets \<Omega> A) \<mu>'"
atLeast_Suc_greaterThan: fixes k :: "nat" shows "{Suc k..} = {k<..}"