hcju/isabelleps huggingface dataset:
size_less_than_select: fixes n :: "nat"   and xs :: "'a list" assumes "n < length xs" shows "size {#y \<in># mset xs. y < select n xs#} \<le> n"
rev_to_bl_eq: fixes w :: "'b word" shows "rev (to_bl w) = map (bit w) [0..<LENGTH('b)]"
IH: fixes x :: "'a" assumes "accessible_on P A x"   and "\<And>y. \<lbrakk>y \<in> A; P y x\<rbrakk> \<Longrightarrow> Q y" shows "Q x"
dres_nres_rel_def: fixes R :: "('a \<times> 'b) set" shows "\<langle>R\<rangle>dres_nres_rel \<equiv> {(c, a). nres_of c \<le> \<Down> R a}"
EncI: fixes F :: "\<Pi>\<^sub>1"   and x :: "\<kappa>"   and v :: "i" assumes "\<exists>r o\<^sub>1. Some r = d\<^sub>1 F \<and> Some o\<^sub>1 = d\<^sub>\<kappa> x \<and> o\<^sub>1 \<in> en r" shows "[\<lbrace>x,F\<rbrace> in v]"
p_irred: shows "\<forall>(E, k)\<in>set P. j < k \<longrightarrow> is_irreducible [C] E"
wens_single_finite_in_wens_set: fixes act :: "('b \<times> 'b) set"   and B :: "'b set"   and k :: "nat"   and init :: "'b set"   and allowed :: "('b \<times> 'b) set set" assumes "single_valued act" shows "wens_single_finite act B k \<in> wens_set (mk_program (init, {act}, allowed)) B"
linear_eq_on: fixes f :: "'d \<Rightarrow> 'e"   and g :: "'d \<Rightarrow> 'e"   and x :: "'d"   and B :: "'d set" assumes "clinear f"   and "clinear g"   and "x \<in> cspan B"   and "\<And>b. b \<in> B \<Longrightarrow> f b = g b" shows "f x = g x"
finvimageD1: fixes a :: "V"   and r :: "V"   and A :: "V" assumes "a \<in>\<^sub>\<circ> r -`\<^sub>\<bullet> A" shows "a \<in>\<^sub>\<circ> \<D>\<^sub>\<bullet> (r \<restriction>\<^sup>r\<^sub>\<bullet> A)"
comp_null_2: fixes f :: "'b" shows "D f B.null = B.null"
kF: shows "\<forall>A\<in>F. finite A \<and> card A \<le> k"
mult_comp_cls_val_process_measurable': fixes Mkt2 :: "('a, 'd) discrete_market"   and pf :: "'d \<Rightarrow> nat \<Rightarrow> 'a \<Rightarrow> real"   and n :: "nat"   and qty :: "nat \<Rightarrow> 'a \<Rightarrow> real" assumes "cls_val_process Mkt2 pf n \<in> borel_measurable (F n)"   and "portfolio pf"   and "qty n \<in> borel_measurable (F n)"   and "0 \<noteq> n" shows "cls_val_process Mkt2 (qty_mult_comp pf qty) n \<in> borel_measurable (F n)"
IH: fixes r :: "'a rexp" shows "slexer r s = lexer r s"
rep_eq: fixes x :: "dflt_size word" shows "Rep_uint (Abs_uint' x) = x"
hv_props_2: shows "\<forall>(k, \<gamma>)\<in>\<V>. \<exists>(k', \<gamma>')\<in>two_chain_vertical_boundary two_chain. \<exists>a\<in>{0..1}. \<exists>b\<in>{0..1}. a \<le> b \<and> subpath a b \<gamma>' = \<gamma>"
tllist_of_llist_eq: fixes b' :: "'b"   and xs :: "'a llist"   and b :: "'b" shows "(tllist_of_llist b' xs = TNil b) = (b = b' \<and> xs = LNil)"
rel_fun_def: fixes A :: "'d \<Rightarrow> 'f \<Rightarrow> bool"   and B :: "'e \<Rightarrow> 'g \<Rightarrow> bool" shows "A ===> B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
wf_action_translation_on_inj_onD: fixes E :: "('a \<times> ('b, 'a) obs_event action) llist"   and E' :: "('a \<times> ('b, 'a) obs_event action) llist"   and A :: "nat set"   and f :: "nat \<Rightarrow> nat" assumes "wf_action_translation_on E E' A f" shows "inj_on f (actions E)"
dirichlet_prod_altdef1: fixes f :: "nat \<Rightarrow> 'b"   and g :: "nat \<Rightarrow> 'b" shows "f \<star> g = (\<lambda>n. \<Sum>d | d dvd n. f (n div d) * g d)"
fmap_of_list_simps_1: shows "fmap_of_list [] = fmempty"
reduced_GB_nonzero_dgrad_p_set: fixes d :: "'a \<Rightarrow> nat"   and F :: "('a \<Rightarrow>\<^sub>0 'd) set"   and m :: "nat" assumes "dickson_grading d"   and "finite (local.punit.component_of_term ` Keys F)"   and "F \<subseteq> punit.dgrad_p_set d m" shows "0 \<notin> punit.reduced_GB F"
isomorphic_symmetric: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "VoV.isomorphic f g" shows "VoV.isomorphic g f"
assms: shows "SubstForm v \<lbrakk>quot_dbtm u\<rbrakk>e x x'"
ivlflows_def: fixes stops :: "'c sctn set"   and stopcont :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set \<Rightarrow> ('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set \<times> ('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set"   and trap :: "('c \<times> 'c \<Rightarrow>\<^sub>L 'c) set"   and rsctn :: "'c sctn" shows "ivlflows stops stopcont trap rsctn = (\<forall>ivl\<subseteq>\<Union> (plane_of ` stops) \<times> UNIV. ivl \<subseteq> snd (stopcont ivl) \<and> fst (stopcont ivl) \<subseteq> snd (stopcont ivl) \<and> fst (stopcont ivl) \<subseteq> sbelow_halfspace rsctn \<times> UNIV \<and> snd (stopcont ivl) \<subseteq> sbelow_halfspace rsctn \<times> UNIV \<and> flowsto ivl {0..} (snd (stopcont ivl)) (fst (stopcont ivl) \<union> trap))"
True: fixes F :: "('x \<Rightarrow> bool) \<Rightarrow> bool" assumes "is_filter F" shows "F (\<lambda>x. True)"
degree_m_eq: fixes f :: "int poly" assumes "lead_coeff f mod p ^ n \<noteq> 0"   and "1 < p ^ n" shows "degree_m f = degree f"
Inl_iff: fixes a :: "hf"   and b :: "hf" shows "(HF.Inl a = HF.Inl b) = (a = b)"
nv: shows "norm1 (\<chi>i. cmod (local.max_v $h i)) = 1"
max_fun_diff: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'a" assumes "f \<noteq> g"   and "f \<in> rs.FINFUNC"   and "g \<in> rs.FINFUNC" shows "\<exists>a b. a \<noteq> b \<and> a \<in> Field r \<and> b \<in> Field r \<and> f (s.max_fun_diff f g) = a \<and> g (s.max_fun_diff f g) = b"
Int_Union: fixes A :: "'a set"   and B :: "'a set set" shows "A \<inter> \<Union> B = \<Union> ((\<inter>) A ` B)"
dp_aux3: fixes z :: "'a"   and w :: "'a"   and x :: "'a"   and y :: "'a" assumes "is_p_fun z"   and "is_total w"   and "x\<^sup>\<smile> ; z = 1" shows "(y ; z\<^sup>\<smile> \<cdot> w ; x\<^sup>\<smile>) ; z = y"
ex_countable_basis: shows "\<exists>B. countable B \<and> topological_basis B"
61101: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "- (- - x \<sqinter> y) \<squnion> z = - x \<squnion> (- y \<squnion> z)"
poly_mult_normalize: fixes p1 :: "'a list"   and p2 :: "'a list" assumes "set p1 \<subseteq> carrier R"   and "set p2 \<subseteq> carrier R" shows "poly_mult p1 p2 = poly_mult (local.normalize p1) p2"
abs_metric_completion_def: shows "abs_metric_completion \<equiv> quot_type.abs (\<lambda>u v. Cauchy u \<and> Cauchy v \<and> (\<lambda>n. dist (u n) (v n)) \<longlonglongrightarrow> 0) Abs_metric_completion"
ncoplanar_perm_19: 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 "\<not> Tarski_neutral_dimensionless.Coplanar Bet A B C D" shows "\<not> Tarski_neutral_dimensionless.Coplanar Bet D A C B"
pderivs_lang_Star: fixes r :: "'a rexp" shows "pderivs_lang UNIV1 (Star r) \<subseteq> Timess (pderivs_lang UNIV1 r) (Star r)"
cbox: fixes u :: "'a"   and v :: "'a" assumes "cbox u v \<in> \<D>" shows "\<exists>n. \<forall>i\<in>Basis. v \<bullet> i - u \<bullet> i = (b \<bullet> i - a \<bullet> i) / 2 ^ n"
gcd_neg1: fixes one :: "'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and gcd :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and lcm :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and unit_factor :: "'a \<Rightarrow> 'a"   and normalize :: "'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.ring_gcd one zero times gcd lcm divide plus minus unit_factor normalize uminus" shows "gcd (uminus a) b = gcd a b"
zeroD: fixes x :: "'c"   and u :: "'c \<Rightarrow> real" assumes "zero x ([u]\<^sub>\<R>)"   and "u \<in> V" shows "u x = 0"
assms_1: shows "nonneg_cexpr (shift_var_set V) (case_nat t \<Gamma>) e"
CallThrowParams: fixes es :: "expr list"   and vs :: "val list"   and r :: "nat \<times> char list list"   and es' :: "expr list"   and P :: "(char list \<times> base list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> expr) list) list"   and E :: "char list \<Rightarrow> ty option"   and v :: "val"   and Copt :: "char list option"   and M :: "char list"   and s :: "(nat \<Rightarrow> (char list \<times> (char list list \<times> (char list \<Rightarrow> val option)) set) option) \<times> (char list \<Rightarrow> val option)" assumes "es = map Val vs @ Throw r # es'" shows "redp P E (Call (Val v) Copt M es) s (Throw r) s"
transformation_group_closed: fixes G :: "('c \<Rightarrow> 'c) set"   and S :: "'c set"   and \<alpha> :: "'c \<Rightarrow> 'c"   and x :: "'c" assumes "transformation_group G S"   and "\<alpha> \<in> G"   and "x \<in> S" shows "\<alpha> x \<in> S"
diamond_eq_characterisations: fixes x :: "real"   and y :: "real" shows "diamond_cube (x, y) = diamond_cube_gen (x, y)"
subcls1_acyclic: assumes "ws_prog" shows "acyclic subcls1"
encode1_lt: fixes i :: "nat" assumes "i < d" shows "encode1 i < d1"
FL_valid_Act: fixes P :: "'state"   and f :: "'effect"   and \<alpha> :: "'act"   and x :: "('b, 'pred, 'act, 'effect) formula" shows "P \<Turnstile> Act f \<alpha> x = (\<exists>\<alpha>'' x' P'. Act f \<alpha> x = Act f \<alpha>'' x' \<and> \<langle>f\<rangle>P \<rightarrow> \<langle>\<alpha>'',P'\<rangle> \<and> P' \<Turnstile> x')"
restrict_below_Rep_eq: fixes x :: "'a"   and y :: "'a"   and z :: "'a" assumes "x \<sqinter> - - y \<le> z" shows "x \<sqinter> y = x \<sqinter> z \<sqinter> y"
h1: shows "invertible_mat (M_mat (signs_smash signs1 signs2) (subsets_smash (length qs1) subsets1 subsets2))"
cobounded2: fixes a :: "'k"   and b :: "'k" shows "min a b \<le> b"
inject: fixes x21 :: "'b"   and x22 :: "'b"   and y21 :: "'b"   and y22 :: "'b" shows "(Point x21 x22 = Point y21 y22) = (x21 = y21 \<and> x22 = y22)"
line_extension_in_carrier: fixes K :: "'a set"   and a :: "'a"   and E :: "'a set" assumes "K \<subseteq> carrier R"   and "a \<in> carrier R"   and "E \<subseteq> carrier R" shows "line_extension K a E \<subseteq> carrier R"
filter_False: fixes xs :: "'g list"   and P :: "'g \<Rightarrow> bool" assumes "\<forall>x\<in>set xs. \<not> P x" shows "filter P xs = []"
assms: fixes \<sigma> :: "'a \<Rightarrow> 'b"   and \<sigma>' :: "'a \<Rightarrow> 'b"   and j :: "'a" assumes "U \<sigma> \<sigma>'"   and "j \<in> J" shows "\<sigma>' j = \<sigma> j"
card_UNIV_finfun: shows "CARD(('a, 'b) finfun) = (if CARD('a) \<noteq> 0 \<and> CARD('b) \<noteq> 0 \<or> CARD('b) = 1 then CARD('b) ^ CARD('a) else 0)"
child_single_level: fixes d :: "nat"   and p :: "(nat \<times> int) list"   and dir :: "dir" assumes "d < length p" shows "lv p d < lv (child p dir d) d"
remdups_adj_append_dropWhile: fixes xs :: "'b list"   and y :: "'b"   and ys :: "'b list" shows "remdups_adj (xs @ y # ys) = remdups_adj (xs @ [y]) @ remdups_adj (dropWhile (\<lambda>x. x = y) ys)"
f2: shows "(r |\<^bsub>rel\<^esub> s) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (t |\<^bsub>rel\<^esub> u) \<otimes>\<^bsub>rec_monoid_rng_of_frac\<^esub> (v |\<^bsub>rel\<^esub> w) = (r \<otimes> t \<otimes> v |\<^bsub>rel\<^esub> s \<otimes> u \<otimes> w)"
borel_measurable_max: fixes f :: "'b \<Rightarrow> 'c"   and M :: "'b measure"   and g :: "'b \<Rightarrow> 'c" assumes "f \<in> borel_measurable M"   and "g \<in> borel_measurable M" shows "(\<lambda>x. max (g x) (f x)) \<in> borel_measurable M"
int_pow_one: fixes z :: "int" shows "[z] \<cdot> \<zero> = \<zero>"
cr: shows "lookup r (t \<oplus> lt f) = - lookup p (t \<oplus> lt f)"
Ord_odiff_eq: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "\<alpha> \<subseteq>\<^sub>\<circ> \<beta>"   and "Ord \<alpha>"   and "Ord \<beta>" shows "\<alpha> + odiff \<beta> \<alpha> = \<beta>"
inner_add_right: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and x :: "'b"   and y :: "'b"   and z :: "'b" assumes "inner_product_space scale inner" shows "inner x (y + z) = inner x y + inner x z"
af_append_true: fixes \<phi> :: "'a ltln"   and w :: "'a set list"   and w' :: "'a set list" assumes "af \<phi> w \<sim> true\<^sub>n" shows "af \<phi> (w @ w') \<sim> true\<^sub>n"
eval_cases_9: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and e\<^sub>1 :: "char list exp"   and e\<^sub>2 :: "char list exp"   and s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and e' :: "char list exp"   and s' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>e\<^sub>1;; e\<^sub>2,s\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>" obtains v :: "val" and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" where "P \<turnstile> \<langle>e\<^sub>1,s\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1\<rangle>" and "P \<turnstile> \<langle>e\<^sub>2,s\<^sub>1\<rangle> \<Rightarrow> \<langle>e',s'\<rangle>"   | e :: "char list exp" where "e' = throw e" and "P \<turnstile> \<langle>e\<^sub>1,s\<rangle> \<Rightarrow> \<langle>throw e,s'\<rangle>"
m_weakly_prefers_Pm: fixes Pm :: "'a \<Rightarrow> ('b \<times> 'b) set"   and Pw :: "'b \<Rightarrow> ('a \<times> 'a) set"   and \<mu> :: "('a \<times> 'b) set"   and m :: "'a"   and w :: "'b"   and w' :: "'b" assumes "StableMarriage Pm Pw"   and "StableMarriage.match Pm Pw \<mu>"   and "(m, w) \<in> \<mu>" shows "(w' \<in> StableMarriage.m_weakly_prefers Pm m \<mu>) = ((w, w') \<in> Pm m)"
split: fixes P :: "'b \<Rightarrow> bool"   and f1 :: "'b"   and f2 :: "'c \<Rightarrow> 'b"   and f3 :: "'b"   and op :: "'c op" shows "P (case op of Empty \<Rightarrow> f1 | Enq x \<Rightarrow> f2 x | Deq \<Rightarrow> f3) = ((op = Empty \<longrightarrow> P f1) \<and> (\<forall>x2. op = Enq x2 \<longrightarrow> P (f2 x2)) \<and> (op = Deq \<longrightarrow> P f3))"
in_set_lookup_of_listD: fixes v :: "'a"   and f :: "'a \<Rightarrow> 'b list"   and vs :: "'a list"   and xs :: "'b list" assumes "v \<in> set (lookup_tries (tries_of_list f vs) xs)" shows "v \<in> set vs"
chamber_system_def: shows "codomain.\<C> \<equiv> Collect codomain.chamber"
add_diff_swap: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "a \<in> E"   and "b \<in> E"   and "c \<in> E"   and "d \<in> E"   and "a + b = c + d" shows "a - c = d - b"
intro: fixes D :: "'a set"   and cmp :: "'a \<Rightarrow> 'a \<Rightarrow> comp_res" assumes "linorder_on D cmp"   and "eq_linorder_on_axioms D cmp" shows "eq_linorder_on D cmp"
assign: fixes e :: "'exp"   and m :: "'id \<Rightarrow> 'val"   and v :: "'val"   and x :: "'id"   and \<iota> :: "nat" assumes "E e m = v" shows "\<langle>x:=\<^bsub>\<iota>\<^esub> e,m\<rangle> \<rightarrow>\<lhd>[]\<rhd> \<langle>None,m(x := v)\<rangle>"
mult_poly_0_right: fixes p :: "'c poly" shows "p * 0 = 0"
s: shows "\<forall>n. s n \<in> \<omega>_limit_set x - range (flow0 y) \<and> infdist (s n) (range (flow0 y)) < infdist p (range (flow0 y)) / 2 ^ n"
SPLIT_def: fixes v :: "'b"   and w :: "'b" shows "SPLIT v w \<equiv> v = w"
congruent_triangleI_sss: fixes a1 :: "'b"   and b1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c1 :: "'b"   and c2 :: "'c" assumes "dist a1 b1 = dist a2 b2"   and "dist b1 c1 = dist b2 c2"   and "dist a1 c1 = dist a2 c2" shows "congruent_triangle a1 b1 c1 a2 b2 c2"
children_eq_h3: fixes ptr' :: "(_) object_ptr"   and children :: "(_) node_ptr list" assumes "ptr \<noteq> ptr'" shows "h3 \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children = h' \<turnstile> get_child_nodes ptr' \<rightarrow>\<^sub>r children"
lm007: fixes X :: "'c set"   and P :: "('c \<times> 'd) set"   and Q :: "('c \<times> 'd) set" assumes "X \<inter> Domain P \<subseteq> Domain Q" shows "P +< Q = (P outside X) +< Q"
rep_eq: fixes x :: "'d \<Rightarrow> 'c"   and xa :: "'d cset" shows "rcset (x `\<^sub>c xa) = x ` rcset xa"
measure_le_1: fixes M :: "'c measure"   and X :: "'c set" assumes "prob_space M" shows "emeasure M X \<le> 1"
Actual_out_SDG_Def: fixes n :: "'node SDG_node"   and m :: "'node"   and x :: "nat"   and V :: "'var" assumes "valid_SDG_node n"   and "n = Actual_out (m, x)"   and "V = ParamDefs m ! x" shows "V \<in> Def\<^bsub>SDG\<^esub> n"
x_c_nc_split: fixes x :: "'a" shows "x \<cdot> 1\<^sub>\<pi> \<parallel> nc = (x \<sqinter> nc) \<cdot> nc + x \<cdot> (0::'a) \<parallel> nc"
abc_append_first_step_eq_pre: fixes st :: "nat"   and A :: "abc_inst list"   and lm :: "nat list"   and B :: "abc_inst list" assumes "st < length A" shows "abc_step_l (st, lm) (abc_fetch st (A @ B)) = abc_step_l (st, lm) (abc_fetch st A)"
stat_acc: shows "G \<turnstile> m of statC accessible_from accC"
normalize_gcd_right: fixes a :: "'a"   and b :: "'a" shows "gcd a (normalize b) = gcd a b"
IfE: fixes \<pi> :: "char list \<Rightarrow> com option"   and b :: "bexp"   and c1 :: "com"   and c2 :: "com"   and s :: "char list \<Rightarrow> int \<Rightarrow> int"   and t :: "char list \<Rightarrow> int \<Rightarrow> int" assumes "\<pi>: (IF b THEN c1 ELSE c2, s) \<Rightarrow> t" obtains "bval b s" and "\<pi>: (c1, s) \<Rightarrow> t"   | "\<not> bval b s" and "\<pi>: (c2, s) \<Rightarrow> t"
CS_unfold: fixes c :: "com" shows "CS c = step UNIV (CS c)"
volume_def: fixes M :: "'b set set"   and f :: "'b set \<Rightarrow> ennreal" shows "volume M f = (f {} = 0 \<and> (\<forall>a\<in>M. 0 \<le> f a) \<and> (\<forall>C\<subseteq>M. disjoint C \<longrightarrow> finite C \<longrightarrow> \<Union> C \<in> M \<longrightarrow> f (\<Union> C) = sum f C))"
EQ: fixes a :: "'a" assumes "a \<in> Field r" shows "f a = g a"
lm099: fixes Y :: "'d set" shows "injections {} Y = {{}}"
invc_baldL: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and a :: "'b" assumes "invc2 l"   and "invc r"   and "color r = Black" shows "invc (baldL l a r)"
hyps: fixes s :: "'a list"   and v :: "'a val" assumes "s \<in> deriv c r \<rightarrow> v" shows "(c # s) \<in> r \<rightarrow> injval r c v"
pnormal_last_nonzero: fixes p :: "'a list" assumes "pnormal p" shows "last p \<noteq> (0::'a)"
module_hom_add: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c" assumes "module_hom s1 s2 f"   and "module_hom s1 s2 g" shows "module_hom s1 s2 (\<lambda>x. f x + g x)"
strict_mono_on_imp_inj_on: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set" assumes "Fun.strict_mono_on f A" shows "inj_on f A"
Sup_downset_adj: shows "Sup \<stileturn> \<down>"
Basis_le_norm: fixes b :: "'b"   and x :: "'b" assumes "b \<in> Basis" shows "\<bar>x \<bullet> b\<bar> \<le> norm x"
A: shows "A \<in> carrier_mat (dim_row A) (dim_row A)"
sum_Suc: fixes f :: "'b \<Rightarrow> nat"   and A :: "'b set" shows "(\<Sum>x\<in>A. Suc (f x)) = sum f A + card A"
close_sub: fixes \<Gamma> :: "'a list"   and \<phi> :: "'a" shows "sublist (close \<Gamma> \<phi>) \<Gamma>"
assms_3: fixes n :: "nat" shows "g n \<in> S"
eval_val_not_bot: fixes P :: "db"   and V :: "db"   and \<rho> :: "nat \<rightarrow> ValD" assumes "P \<Down> V" shows "evalDdb V\<cdot>\<rho> \<noteq> \<bottom>"
ide_REP: fixes V :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a> :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i> :: "'b \<Rightarrow> 'b"   and src :: "'b \<Rightarrow> 'b"   and trg :: "'b \<Rightarrow> 'b"   and A :: "('b, 'b set) Maps.arr" assumes "maps_category V H \<a> \<i> src trg"   and "partial_magma.arr (concrete_category.COMP (Collect (horizontal_homs.obj V src)) (\<lambda>a b. {F. \<exists>f. horizontal_homs.in_hhom V src trg f a b \<and> (\<exists>g. bicategory.adjoint_pair V H \<a> \<i> src trg f g) \<and> F = category.iso_class V f}) (\<lambda>_ _ _. maps_category.Comp V H)) A" shows "partial_magma.ide V (maps_category.REP V H \<a> \<i> src trg A)"
lbound_ofD1: fixes x :: "'a"   and y :: "'a"   and b :: "'a" assumes "dual_order.lbound_of x y b" shows "order.greater_eq b x"
is_elementary_monoidal_category: shows "elementary_monoidal_category (\<cdot>) (\<otimes>) \<I> lunit\<^sub>F\<^sub>M\<^sub>C runit\<^sub>F\<^sub>M\<^sub>C assoc\<^sub>F\<^sub>M\<^sub>C"
prem: fixes n :: "enat" assumes "\<forall>m<n. P m" shows "P n"
a_supdist_var: fixes x :: "'a"   and y :: "'a" shows "ad (x + y) \<le> ad x"
sAsA': fixes n :: "nat" assumes "n < length paid_A'" shows "s_A' n = s_A n"
subdegree_diff_eq1: fixes f :: "'a fps"   and g :: "'a fps" assumes "f \<noteq> 0"   and "subdegree f < subdegree g" shows "subdegree (f - g) = subdegree f"
l9_31: fixes A :: "'p"   and X :: "'p"   and Y :: "'p"   and Z :: "'p" assumes "A X OS Y Z"   and "A Z OS Y X" shows "A Y TS X Z"
res_e: shows "ord_resolve_rename CAs DA AAs As \<sigma> E"
cases: fixes x :: "'b tree" obtains "x = \<langle>\<rangle>"   | l :: "'b tree" and a :: "'b" and r :: "'b tree" where "x = \<langle>l, a, r\<rangle>"
infinite_countable_subset: fixes S :: "'b set" assumes "infinite S" shows "\<exists>f. inj f \<and> range f \<subseteq> S"
find_sig_reducer_SomeD'_2: fixes bs :: "('t \<times> ('a \<Rightarrow>\<^sub>0 'b)) list"   and u :: "'t"   and t :: "'a"   and i :: "nat"   and j :: "nat"   and b :: "'t \<times> ('a \<Rightarrow>\<^sub>0 'b)" assumes "find_sig_reducer bs u t i = Some j"   and "b = bs ! (j - i)" shows "snd b \<noteq> 0"
Par: fixes R :: "ccs"   and S :: "ccs"   and T :: "ccs"   and U :: "ccs" assumes "(R, S) \<in> Rel"   and "(T, U) \<in> Rel'" shows "(R \<parallel> T, S \<parallel> U) \<in> Rel'"
product_two_squares_aux: fixes a :: "int"   and b :: "int"   and c :: "int"   and d :: "int" shows "sum2sq_int (a, b) * sum2sq_int (c, d) = sum2sq_int (a * c - b * d, a * d + b * c)"
ih: fixes y :: "hmultiset"   and e :: "nat"   and N :: "hmultiset" assumes "y < M"   and "local.well_base\<^sub>h y"   and "local.aligned\<^sub>d e y"   and "local.aligned\<^sub>d e N"   and "y < N" shows "local.decode e y < local.decode e N"
abs_eq: fixes xa :: "'b pmf"   and x :: "'b set" shows "cond_pmf_impl xa x = map_option mapping.Mapping (if x \<inter> set_pmf xa = {} then None else Some (\<lambda>xb. if xb \<in> x \<inter> set_pmf xa then Some (pmf xa xb / measure_pmf.prob xa x) else None))"
SND: shows "((sh, csph + cn), ww, s', csp + (csph + cn)) \<in> trcl (ntrs fg)"
antisym_conv2: fixes x :: "'b list"   and y :: "'b list" assumes "x \<le>p y" shows "(\<not> x <p y) = (x = y)"
base: fixes prb :: "('d, 'e, 'f) pre_RedBlack" assumes "fst (root (red prb)) = init (black prb)"   and "edges (red prb) = {}"   and "subs prb = {}"   and "confs prb (root (red prb)) = init_conf prb"   and "marked prb = (\<lambda>rv. False)"   and "strengthenings prb = (\<lambda>rv \<sigma>. True)" shows "RedBlack prb"
add_mset: fixes x :: "'b"   and N :: "'b multiset" shows "\<Prod>\<^sub># (add_mset x N) = x * \<Prod>\<^sub># N"
Inf_atLeast: fixes x :: "'b \<times> 'c" shows "Inf_prod (prod.atLeast x) = x"
pow_ci: fixes ls :: "'a list"   and n :: "nat"   and P :: "pol" assumes "in_carrier ls" shows "Ipol ls (Commutative_Ring.pow n P) = Ipol ls P [^] n"
satPB_transE: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" assumes "I.wtE \<xi>" shows "I.satPB \<xi> \<Phi>"
not_in_Liminf: shows "C \<notin> Liminf_llist Ns"
cotrace_ldrop: fixes c :: "('a, 'b, 'c) configuration"   and t :: "('a, 'b, 'c) event llist"   and i :: "nat" assumes "cotrace c t"   and "enat i \<le> llength t" shows "cotrace (local.cos c t i) (Co_Snapshot.ldrop i t)"
OrL: fixes F :: "'b formula"   and \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and n :: "nat"   and G :: "'b formula" assumes "F, \<Gamma> \<Rightarrow> \<Delta> \<down> n"   and "G, \<Gamma> \<Rightarrow> \<Delta> \<down> n" shows "F \<^bold>\<or> G, \<Gamma> \<Rightarrow> \<Delta> \<down> Suc n"
dflt_size_by_int: shows "dflt_size = nat_of_integer dflt_size_integer"
Reals_inverse_iff: fixes x :: "'a" shows "(inverse x \<in> \<real>) = (x \<in> \<real>)"
lnths_insert: fixes n :: "nat"   and xs :: "'b llist"   and A :: "nat set" assumes "enat n < llength xs" shows "lselect (insert n A) xs = lselect A (ltake (enat n) xs) $ xs ?! n % lselect {m. Suc (n + m) \<in> A} (ldropn (Suc n) xs)"
prems_5: shows "is_zero_row_upt_k i k A"