hcju/isabelleps huggingface dataset:
sorted_wrt_rev: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a list" shows "sorted_wrt P (rev xs) = sorted_wrt (\<lambda>x y. P y x) xs"
Eq: shows "inputChain xvec M P = inputChain yvec N Q"
simps_2: fixes i :: "'b" shows "nominals (Nom i) = {i}"
supt_size: fixes s :: "('d, 'e) Term.term"   and t :: "('d, 'e) Term.term" assumes "s \<rhd> t" shows "size t < size s"
min_len_of_nat_of: fixes n :: "'b sat" shows "min LENGTH('b) (nat_of n) = nat_of n"
union_lcomm: fixes M :: "'a multiset"   and N :: "'a multiset"   and K :: "'a multiset" shows "M + (N + K) = N + (M + K)"
floor_le_iff: fixes x :: "'a"   and z :: "int" shows "(\<lfloor>x\<rfloor> \<le> z) = (x < of_int z + (1::'a))"
path_path_equals_cycle_aux: fixes x :: "'a"   and y :: "'a" assumes "has_start_end_points_path x"   and "has_start_end_points_path y"   and "start_points x = end_points y"   and "end_points x = start_points y" shows "x \<le> (x + y)\<^sup>T\<^sup>\<star>"
d_def: fixes x :: "'a" shows "d x \<equiv> if x = (0::'a) then 0::'a else x"
intros_4: fixes u :: "'e \<Rightarrow> 'd"   and c :: "'e"   and d :: "'d" assumes "u c = d" shows "u \<turnstile> EQ c d"
insert_rga_None_commutes: fixes i2 :: "'a option"   and e1 :: "'a"   and xs :: "'a list"   and e2 :: "'a" assumes "i2 \<noteq> Some e1" shows "insert_rga (insert_rga xs (e1, None)) (e2, i2) = insert_rga (insert_rga xs (e2, i2)) (e1, None)"
accp_wfPI: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "\<forall>x. Wellfounded.accp r x" shows "wfP r"
lt_plus_distinct_eq_max: fixes p :: "'t \<Rightarrow>\<^sub>0 'd"   and q :: "'t \<Rightarrow>\<^sub>0 'd" assumes "lt p \<noteq> lt q" shows "lt (p + q) = ord_term_lin.max (lt p) (lt q)"
intro: fixes H :: "'c set"   and G :: "'c set"   and composition :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and unit :: "'c" assumes "subgroup_of_group H G composition unit"   and "abelian_group G composition unit" shows "subgroup_of_abelian_group H G composition unit"
col_scalar_prod_as_sum: fixes v :: "'c vec"   and A :: "'c mat"   and j :: "nat" assumes "dim_vec v = dim_row A" shows "col A j \<bullet> v = (\<Sum>i = 0..<dim_vec v. A $$ (i, j) * v $ i)"
geq_imp_mix_geq_right: fixes x :: "'a pmf"   and y :: "'a pmf"   and \<R> :: "('a pmf \<times> 'a pmf) set"   and outcomes :: "'a set"   and \<alpha> :: "real" assumes "x \<succeq>[\<R>] y"   and "rational_preference (\<P> outcomes) \<R>"   and "independent_vnm (\<P> outcomes) \<R>"   and "\<alpha> \<in> {0..1}" shows "mix_pmf \<alpha> x y \<succeq>[\<R>] y"
tss_list_takeWhile: shows "tss_list = takeWhile (\<lambda>t. enat t + right I < enat nt) (linearize tss)"
preserves_cod: fixes f :: "'a" assumes "B.arr f" shows "B.cod (src\<^sub>B f) = src\<^sub>B (B.cod f)"
verit_sum_simplify: fixes a :: "'g" shows "a + (0::'g) = a"
assms_2: shows "S \<subseteq> \<Union> \<F>"
MVT_corrected: fixes D :: "'c set"   and f :: "'c \<Rightarrow> 'd"   and J :: "'c \<Rightarrow> 'c \<Rightarrow> 'd"   and a :: "'c"   and u :: "'c" assumes "\<And>x. x \<in> D \<Longrightarrow> (f has_derivative J x) (at x within D)"   and "\<And>x. \<lbrakk>0 \<le> x; x \<le> 1\<rbrakk> \<Longrightarrow> a + x *\<^sub>R u \<in> D" shows "\<exists>t\<in>Basis \<rightarrow> {0<..<1}. f (a + u) - f a = (\<Sum>i\<in>Basis. (J (a + t i *\<^sub>R u) u \<bullet> i) *\<^sub>R i)"
bij_betw_nsets: fixes f :: "'a \<Rightarrow> 'b"   and A :: "'a set"   and B :: "'b set"   and n :: "nat" assumes "bij_betw f A B" shows "bij_betw ((`) f) ([A]\<^bsup>n\<^esup>) ([B]\<^bsup>n\<^esup>)"
has_sort_empty: fixes \<Sigma> :: "(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)"   and T :: "typ" assumes "wf_sig \<Sigma>"   and "wf_type \<Sigma> T" shows "has_sort (osig \<Sigma>) T full_sort"
adv: assumes "exception_\<I> (\<I>_real2 \<oplus>\<^sub>\<I> \<I>_common2) \<turnstile>g absorb \<A> (obsf_converter (parallel_wiring \<odot> (converter_of_resource real1 |\<^sub>\<propto> 1\<^sub>C))) \<surd>" shows "advantage (absorb \<A> (obsf_converter (parallel_wiring \<odot> (converter_of_resource real1 |\<^sub>\<propto> 1\<^sub>C)))) (obsf_resource (sim2 |\<^sub>= 1\<^sub>C \<rhd> ideal2)) (obsf_resource real2) \<le> adv2"
Un_commute_helper: fixes a :: "'a set"   and b :: "'a set"   and c :: "'a set"   and d :: "'a set" shows "a \<union> b \<union> (c \<union> d) = a \<union> c \<union> (b \<union> d)"
principal_dchar_coprime: fixes k :: "nat"   and n :: "nat" assumes "coprime k n" shows "principal_dchar n k = 1"
sep_conj_commuteI: fixes P :: "'a \<Rightarrow> bool"   and Q :: "'a \<Rightarrow> bool"   and h :: "'a" assumes "(P \<and>* Q) h" shows "(Q \<and>* P) h"
r_inv_ex: fixes x :: "'a vec" assumes "x \<in> carrier_vec n" shows "\<exists>y\<in>carrier_vec n. x + y = 0\<^sub>v n"
power_commuting_commutes: fixes x :: "'a"   and y :: "'a"   and n :: "nat" assumes "x \<cdot> y = y \<cdot> x" shows "x ^ n \<cdot> y = y \<cdot> x ^ n"
has_integral_spike_finite: fixes S :: "'b set"   and T :: "'b set"   and g :: "'b \<Rightarrow> 'c"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "finite S"   and "\<And>x. x \<in> T - S \<Longrightarrow> g x = f x"   and "(f has_integral y) T" shows "(g has_integral y) T"
P1: shows "Bet A B D \<and> Cong B D A B"
nonneg_Reals_mult_I: fixes a :: "'a"   and b :: "'a" assumes "a \<in> \<real>\<^sub>\<ge>\<^sub>0"   and "b \<in> \<real>\<^sub>\<ge>\<^sub>0" shows "a * b \<in> \<real>\<^sub>\<ge>\<^sub>0"
bangActE: fixes \<Psi> :: "'b"   and P :: "('a, 'b, 'c) psi"   and \<alpha> :: "'a action"   and P' :: "('a, 'b, 'c) psi" assumes "\<Psi> \<rhd> !P \<longmapsto> \<alpha> \<prec> P'"   and "bn \<alpha> \<sharp>* subject \<alpha>"   and "guarded P"   and "\<alpha> \<noteq> \<tau>"   and "bn \<alpha> \<sharp>* P" obtains Q :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> P \<longmapsto> \<alpha> \<prec> Q" and "P' \<sim> Q \<parallel> !P"
Gramian_determinant_div: fixes l :: "nat" assumes "l < length fs" shows "d (Suc l) / d l = \<parallel>gso l\<parallel>\<^sup>2"
mult_mod_right: fixes divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and euclidean_size :: "'c \<Rightarrow> nat"   and c :: "'c"   and a :: "'c"   and b :: "'c" assumes "class.euclidean_semiring_cancel divide plus minus zero times one modulo euclidean_size" shows "times c (modulo a b) = modulo (times c a) (times c b)"
simps: fixes a :: "Regions.intv" shows "Regions.isConst a = (\<exists>uu_. a = Regions.intv.Const uu_)"
is_norm_le_1_def: fixes z :: "complex" shows "is_norm_le_1 z = ((Re z)\<^sup>2 + (Im z)\<^sup>2 \<le> 1)"
get_min_member: fixes ts :: "'b tree list" assumes "ts \<noteq> []" shows "get_min ts \<in># mset_heap ts"
that: fixes r :: "V" assumes "x = \<langle>r, \<D>\<^sub>\<circ> r\<rangle>"   and "r \<in>\<^sub>\<circ> A" shows "thesis"
cobounded2: fixes a :: "'g multiset"   and b :: "'g multiset" shows "a \<inter># b \<subseteq># b"
sums: fixes f :: "'b \<Rightarrow> 'c"   and X :: "nat \<Rightarrow> 'b"   and a :: "'b" assumes "bounded_linear f"   and "X sums a" shows "(\<lambda>n. f (X n)) sums f a"
impE: fixes tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and t' :: "nat \<Rightarrow> 'b"   and n :: "nat"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool"   and \<gamma>' :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool" assumes "dynamic_component.eval tCMP active cid t t' n (dynamic_component.imp \<gamma> \<gamma>')" shows "dynamic_component.eval tCMP active cid t t' n \<gamma> \<longrightarrow> dynamic_component.eval tCMP active cid t t' n \<gamma>'"
is_dghm_op: shows "op_dghm \<FF> : op_dg \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> op_dg \<BB>"
with_proj_simps_4: fixes G :: "'a pair_pre_digraph" shows "tail (with_proj G) = fst"
not_neutral_contains_not_neutral: fixes g :: "'c \<Rightarrow> 'a"   and A :: "'c set" assumes "sum_0 g A \<noteq> (0::'a)" obtains a :: "'c" where "a \<in> A" and "g a \<noteq> (0::'a)"
FmToFo: fixes F :: "('d, 'e, 'f, 'g, 'h, 'i, 'j) Functor_scheme"   and X :: "'d"   and Y :: "'e" assumes "PreFunctor F"   and "X \<in> obj\<^bsub>CatDom F\<^esub>"   and "Y \<in> obj\<^bsub>CatCod F\<^esub>"   and "F ## id\<^bsub>CatDom F\<^esub> X = id\<^bsub>CatCod F\<^esub> Y" shows "F @@ X = Y"
a1: shows "F \<in> Filters\<^bsub>T\<^esub>"
goodYyx: shows "qGood (Y #[[y \<and> x]]_xs)"
set_map_of_compr: fixes xs :: "('a \<times> 'b) list" assumes "distinct (map fst xs)" shows "set xs = {(k, v). Map.map_of xs k = Some v}"
L': shows "L = (n0, j0, i0) # (n1, j1, ix1) # L'"
in_mapping_mult_single: fixes x :: "nat"   and x' :: "nat"   and A :: "(nat \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'a" shows "(x \<in> (\<lambda>x. lookup x x') ` keys (A * Var\<^sub>0 x')) = (0 < x \<and> x - 1 \<in> (\<lambda>x. lookup x x') ` keys A)"
heads_eq_0_iff: fixes xs :: "bool list" shows "(heads xs = 0) = (True \<notin> set xs)"
simple_function_lebesgue_if: fixes f :: "'a \<Rightarrow> real"   and S :: "'a set" assumes "simple_function lebesgue f"   and "S \<in> sets lebesgue" shows "simple_function lebesgue (\<lambda>x. if x \<in> S then f x else 0)"
conjugate_id: fixes conjugate :: "'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.conjugate conjugate" shows "conjugate (conjugate a) = a"
v_1: shows "f_a `` vertices (graph_union A B) \<subseteq> vertices G"
F: shows "\<forall>u. u \<notin> S \<longrightarrow> D u = \<infinity>"
\<sigma>_integer: fixes l :: "nat"   and j :: "nat"   and i :: "nat" assumes "l \<le> j"   and "j \<le> i"   and "i < length (map of_int_hom.vec_hom fs)" shows "gs.\<sigma> l i j \<in> \<int>"
infinite: fixes A :: "'b set" assumes "infinite A" shows "card A = 0"
pbij_lens_axioms_def: fixes x :: "'b \<Longrightarrow> 'c" shows "pbij_lens_axioms x \<equiv> \<forall>\<sigma> v \<rho>. put\<^bsub>x\<^esub> \<sigma> v = put\<^bsub>x\<^esub> \<rho> v"
submap_init_submap_exec_i: fixes s1 :: "('c, 'd) fmap"   and s2 :: "('c, 'd) fmap"   and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" assumes "s1 \<subseteq>\<^sub>f s2"   and "sat_precond_as s1 (a # as)" shows "state_succ s1 a \<subseteq>\<^sub>f state_succ s2 a"
arrow_determines_cone_2: fixes d :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr"   and a :: "'c"   and x :: "('j \<Rightarrow> 'c, 'j \<Rightarrow> 'c) concrete_category.arr" assumes "J_C.ide d"   and "arrow_from_functor (\<cdot>) (\<cdot>\<^sub>[\<^sub>J\<^sub>,\<^sub>C\<^sub>]) local.map a d x" shows "J_C.MkArr (constant_functor.map (\<cdot>\<^sub>J) (\<cdot>) a) (J_C.Map d) (J_C.Map x) = x"
inv_rule_incr: fixes T :: "('a, 'b) TS_scheme"   and I :: "'a set"   and J :: "'a set" assumes "init T \<subseteq> I"   and "{I \<inter> J} TS.trans T {> I}"   and "reach T \<subseteq> J" shows "reach T \<subseteq> I"
assms: fixes i :: "nat" assumes "i < max_Var_floatarith fa" shows "s i = i"
distinct_13: fixes x21 :: "'b aexp"   and x22 :: "'b aexp"   and x51 :: "'b gexp"   and x52 :: "'b gexp" shows "Eq x21 x22 \<noteq> Nor x51 x52"
subgroups_of_size_def: fixes p :: "nat" shows "Hgroup.subgroups_of_size p = {Ha. subgroup Ha (G\<lparr>carrier := H\<rparr>) \<and> card Ha = p}"
declassified_proj_GSMP_subset: fixes I :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and A :: "('lbl strand_label \<times> ('fun, 'var) strand_step) list"   and n :: "'lbl" assumes "wt\<^sub>s\<^sub>u\<^sub>b\<^sub>s\<^sub>t I"   and "wf\<^sub>t\<^sub>r\<^sub>m\<^sub>s (subst_range I)"   and "subst_domain I = \<V> \<and> ground (subst_range I)" shows "declassified\<^sub>l\<^sub>s\<^sub>t A I \<subseteq> GSMP (trms\<^sub>l\<^sub>s\<^sub>t (proj n A))"
aezfun_common_supp_spanning_set': fixes A :: "'a set" assumes "finite A" shows "\<exists>as. distinct as \<and> set as = A \<and> (\<forall>f. supp f \<subseteq> A \<longrightarrow> (\<exists>bs. length bs = length as \<and> f = sum_list (map2 (\<delta>) bs as)))"
abs_ge_minus_self: fixes abs :: "'c \<Rightarrow> 'c"   and plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and uminus :: "'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.ordered_ab_group_add_abs abs plus zero minus uminus less_eq less" shows "less_eq (uminus a) (abs a)"
hd_Cons_tl: fixes xs :: "'c list" assumes "xs \<noteq> []" shows "hd xs # tl xs = xs"
a_inf_complement_bot: fixes x :: "'a" shows "- x \<sqinter> x = bot"
simps_2: fixes \<eta> :: "trm option" shows "Geqo undeft \<eta> = undeff"
length'_def: fixes as :: "'a iarray" shows "IArray.length' as = integer_of_nat (length (IArray.list_of as))"
listDiffIff: fixes x :: "'b"   and a :: "'b list"   and b :: "'b list" shows "(x \<in> set a \<and> x \<notin> set b) = (x \<in> set (list_diff a b))"
length_foldr_update_matchF_step: fixes I :: "\<I>"   and mr :: "mregex"   and mrs :: "mregex list"   and nt :: "nat"   and aux :: "(nat \<times> 'a option list set list \<times> 'a option list set) list"   and base :: "(nat \<times> 'a option list set list \<times> 'a option list set) list \<times> (mregex, 'a option list set) mapping" shows "length (fst (foldr (update_matchF_step I mr mrs nt) aux base)) = length aux + length (fst base)"
pr0_def: fixes a :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr"   and b :: "(hf set, hf \<Rightarrow> hf) concrete_category.arr" shows "hfsetcat.pr0 a b = (if ide a \<and> ide b then mkArr (local.set (hfsetcat.prod a b)) (local.set b) (\<lambda>x. UP (hsnd (DOWN x))) else null)"
range_Some: fixes A :: "'a set"   and a :: "'a" assumes "MaxR_opt A = Some a" shows "a \<in> A \<inter> Field r"
maxi_last: fixes xs :: "'c list"   and key :: "'c \<Rightarrow> 'd"   and i :: "nat" assumes "xs \<noteq> []"   and "maxi xs key < i"   and "i < length xs" shows "key (xs ! i) < key (xs ! maxi xs key)"
eval_constraint_sat_eq: fixes n :: "nat"   and A :: "nat set"   and x :: "event_data option list"   and t1 :: "Formula.trm"   and t2 :: "Formula.trm"   and p :: "bool"   and c :: "mconstraint"   and \<sigma> :: "(String.literal \<times> event_data list) trace"   and V :: "String.literal \<Rightarrow> (nat \<Rightarrow> event_data list set) option"   and i :: "nat" assumes "wf_tuple n A x"   and "fv_trm t1 \<subseteq> A"   and "fv_trm t2 \<subseteq> A"   and "\<forall>i\<in>A. i < n" shows "eval_constraint (t1, p, c, t2) x = Formula.sat \<sigma> V (map the x) i (formula_of_constraint (t1, p, c, t2))"
PLam_ensures: fixes i :: "nat"   and I :: "nat set"   and F :: "nat \<Rightarrow> ('c \<times> (nat \<Rightarrow> 'c) \<times> 'd) program"   and A :: "'c set"   and B :: "'c set" assumes "i \<in> I"   and "F i \<in> A \<times> UNIV ensures B \<times> UNIV"   and "\<forall>j. F j \<in> preserves snd" shows "PLam I F \<in> lift_set i (A \<times> UNIV) ensures lift_set i (B \<times> UNIV)"
RestrictedP_Mem: fixes s :: "tm"   and k :: "tm"   and s' :: "tm"   and a :: "tm"   and b :: "tm" shows "{RestrictedP s k s', HPair a b IN s, a IN k} \<turnstile> HPair a b IN s'"
init_obj_predE: fixes x :: "(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 xa :: "char list"   and xb :: "char list list \<times> (char list, val) mapping" assumes "pred.eval (init_obj_pred x xa) xb" obtains "init_obj' x xa xb"
AE_streams: fixes s :: "'s" assumes "s \<in> S" shows "AE \<omega> in K.T s. \<omega> \<in> streams S"
Nml_Nmlize_2: fixes V :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and src :: "'a \<Rightarrow> 'a"   and trg :: "'a \<Rightarrow> 'a"   and t :: "'a local.term" assumes "bicategorical_language V src trg"   and "bicategorical_language.Arr V src trg t" shows "bicategorical_language.Src src (bicategorical_language.Nmlize V t) = bicategorical_language.Src src t"
iso_locally_ringed_spaces_axioms: shows "iso_locally_ringed_spaces X is_open \<O>\<^sub>X \<rho> b add_str mult_str zero_str one_str Spec is_zariski_open sheaf_spec sheaf_spec_morphisms \<O>b add_sheaf_spec mult_sheaf_spec zero_sheaf_spec one_sheaf_spec f \<phi>\<^sub>f"
smult_one: fixes x :: "'a vec" assumes "x \<in> carrier_vec (n + 1)" shows "(1::'a) \<cdot>\<^sub>v x = x"
subset_singletonD: fixes A :: "'a set"   and x :: "'a" assumes "A \<subseteq> {x}" shows "A = {} \<or> A = {x}"
simps_2: fixes n :: "nat"   and w :: "'a \<times> bool list"   and ws :: "('a \<times> bool list) list" shows "wf_word n (w # ws) = (w \<in> (set \<circ> \<sigma> \<Sigma>) n \<and> wf_word n ws)"
preserves_hom: fixes f :: "'a"   and a :: "'a"   and b :: "'a" assumes "\<guillemotleft>f : a \<Rightarrow> b\<guillemotright>" shows "\<guillemotleft>R f : R a \<Rightarrow> R b\<guillemotright>"
pos_even: fixes n :: "int" assumes "0 \<le> n" shows "even (int_option_enumeration (Some n)) \<and> 0 < int_option_enumeration (Some n)"
not_unifiable_imp_not_empty_NF: fixes E :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) multiset" assumes "\<not> unifiable (set_mset E)" shows "\<exists>E'. E' \<noteq> {#} \<and> (E, E') \<in> unif\<^sup>!"
wf_eq_check_append': fixes V :: "'c set"   and S :: "('d, 'c) strand_step list"   and S' :: "('d, 'c) strand_step list"   and t :: "('d, 'c) Term.term"   and t' :: "('d, 'c) Term.term" assumes "wf\<^sub>s\<^sub>t V (S @ S')" shows "wf\<^sub>s\<^sub>t V (S @ \<langle>check: t \<doteq> t'\<rangle>\<^sub>s\<^sub>t # S')"
inpow_subset_nodes: fixes xs :: "nat list" assumes "xs \<in> (rev \<circ> sorted_list_of_set) ` Pow (set nodes)" shows "set xs \<subseteq> set nodes"
extendContain: fixes r :: "sequent list \<times> sequent"   and ps :: "sequent list"   and c :: "sequent"   and Ps :: "sequent list"   and C :: "sequent"   and S :: "sequent"   and p :: "sequent" assumes "r = (ps, c)"   and "(Ps, C) = extendRule S r"   and "p \<in> set ps" shows "extend S p \<in> set Ps"
CS_def: fixes c :: "com" shows "CS c = Collecting.lfp c (step UNIV)"
rtrancl_is_final: fixes r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c"   and b :: "'c" assumes "r\<^sup>*\<^sup>* a b"   and "relation.is_final r a" shows "a = b"
fgw: fixes w :: "'b" assumes "v \<noteq> w" shows "f w = g w"
additive_def: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'b \<Rightarrow> 'b" shows "sup.additive sup f \<equiv> \<forall>x y. f (sup x y) = sup (f x) (f y)"
upred_ref_iff: fixes Q :: "'b upred"   and P :: "'b upred" shows "(P \<sqsubseteq> Q) = (\<forall>b. \<lbrakk>Q\<rbrakk>\<^sub>e b \<longrightarrow> \<lbrakk>P\<rbrakk>\<^sub>e b)"
ker_altdef: shows "ker = {a \<in> {1..<n}. fermat_liar a n}"
finite_real_of_nat_le_real: fixes u :: "real" shows "finite {n. real n \<le> u}"
map_scale_eq_0_iff: fixes c :: "'c"   and t :: "'b \<Rightarrow>\<^sub>0 'c" shows "(c \<cdot> t = 0) = (c = (0::'c) \<or> t = 0)"
bi_uniqueI: fixes R :: "'b \<Rightarrow> 'c \<Rightarrow> bool" assumes "left_unique R"   and "right_unique R" shows "bi_unique R"
le_lfp: fixes f :: "'a \<Rightarrow> 'a"   and x :: "'a" assumes "mono f"   and "x \<le> lfp f" shows "f x \<le> lfp f"
orthogonal_proj_set: fixes scale :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and inner :: "'b \<Rightarrow> 'b \<Rightarrow> 'a"   and y :: "'b"   and C :: "'b set"   and a :: "'b" assumes "inner_product_space scale inner"   and "y \<in> C"   and "finite C"   and "pairwise (inner_product_space.orthogonal inner) C" shows "inner_product_space.orthogonal inner (a - inner_product_space.proj_onto scale inner a C) y"
least_field_is_type: fixes 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 C :: "char list"   and F :: "char list"   and T :: "ty"   and Cs :: "char list list"   and wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "P \<turnstile> C has least F:T via Cs"   and "wf_prog wf_md P" shows "is_type P T"
less_pratt1_def: fixes x :: "pratt1"   and y :: "pratt1" shows "x < y \<equiv> x \<le> y \<and> x \<noteq> y"
sstart_eq: fixes x :: "'a stream"   and S :: "'a set"   and xs :: "'a list" assumes "x \<in> streams S" shows "(x \<in> sstart S xs) = (\<forall>i<length xs. x !! i = xs ! i)"
is_elementary_monoidal_category: fixes C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "free_monoidal_category C" shows "elementary_monoidal_category (free_monoidal_category.comp C) (free_monoidal_category.tensor\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.unity\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.lunit\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.runit\<^sub>F\<^sub>M\<^sub>C C) (free_monoidal_category.assoc\<^sub>F\<^sub>M\<^sub>C C)"
walk_iarray'_Cons: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'b"   and a :: "'c"   and xs :: "'c list"   and x :: "'b"   and l :: "nat"   and p :: "nat" shows "walk_iarray' f (IArray (a # xs)) x l (Suc p) = walk_iarray' f (IArray xs) x l p"
assms: fixes x :: "'a"   and a\<^sub>0 :: "'b"   and a\<^sub>1 :: "'b" shows "f\<^sub>1 x a\<^sub>0 a\<^sub>1 \<longrightarrow> f\<^sub>2 x a\<^sub>0 a\<^sub>1"
similar_mat_wit_four_block: fixes A1 :: "'b mat"   and B1 :: "'b mat"   and P1 :: "'b mat"   and Q1 :: "'b mat"   and A2 :: "'b mat"   and B2 :: "'b mat"   and P2 :: "'b mat"   and Q2 :: "'b mat"   and URA :: "'b mat"   and UR :: "'b mat"   and LLA :: "'b mat"   and LL :: "'b mat"   and n :: "nat"   and m :: "nat" assumes "similar_mat_wit A1 B1 P1 Q1"   and "similar_mat_wit A2 B2 P2 Q2"   and "URA = P1 * UR * Q2"   and "LLA = P2 * LL * Q1"   and "A1 \<in> carrier_mat n n"   and "A2 \<in> carrier_mat m m"   and "LL \<in> carrier_mat m n"   and "UR \<in> carrier_mat n m" shows "similar_mat_wit (four_block_mat A1 URA LLA A2) (four_block_mat B1 UR LL B2) (four_block_mat P1 (0\<^sub>m n m) (0\<^sub>m m n) P2) (four_block_mat Q1 (0\<^sub>m n m) (0\<^sub>m m n) Q2)"
abs_eq: fixes xa :: "'b \<Rightarrow> 'd \<Rightarrow> 'c"   and x :: "'b \<Rightarrow>\<^sub>0 'd" shows "mapp xa x = Abs_poly_mapping (\<lambda>k. if k \<in> keys x then xa k (lookup x k) else (0::'c))"
node1: fixes i :: "nat"   and R :: "nat set" shows "\<langle>i : onp i : R\<rangle>\<^sub>o \<Turnstile>\<^sub>A (\<lambda>\<sigma> _. oarrivemsg I \<sigma>, other U {i} \<rightarrow>) globala (\<lambda>(\<sigma>, a, uu_). castmsg (I \<sigma>) a)"
lookup_numeral: fixes n :: "num"   and k :: "'c" shows "lookup (numeral n) k = (numeral n when k = (0::'c))"
normal_valuation_belonging_to_prime_divisor_def: fixes K :: "('c, 'd) Ring_scheme"   and P :: "('c \<Rightarrow> ant) set" shows "\<nu>\<^bsub>K P\<^esub> = n_val K (SOME v. v \<in> P)"
fps_of_poly_code: fixes p :: "'b poly" shows "fps_of_poly p = fps_of_ratfps (ratfps_of_poly p)"
maximal_existence_flow_1: fixes x :: "real \<Rightarrow> 'a"   and K :: "real set"   and t0 :: "real"   and x0 :: "'a" assumes "(x solves_ode f) K X"   and "x t0 = x0"   and "is_interval K"   and "t0 \<in> K"   and "K \<subseteq> T" shows "K \<subseteq> existence_ivl t0 x0"
Int_left_absorb: fixes A :: "'e set"   and B :: "'e set" shows "A \<inter> (A \<inter> B) = A \<inter> B"
ucast_eq: fixes w :: "'d word" shows "ucast w = word_of_int (uint w)"
star_trans: fixes x :: "('a \<times> 'a) set" shows "x\<^sup>* O x\<^sup>* \<subseteq> x\<^sup>*"
inverse_Real: fixes X :: "nat \<Rightarrow> rat" assumes "cauchy X" shows "inverse (Real X) = (if vanishes X then 0 else Real (\<lambda>n. inverse (X n)))"
diff_zero: fixes a :: "'k" shows "a - (0::'k) = a"
defs_finite: fixes g :: "'a"   and n :: "'b" shows "finite (defs g n)"
5: shows "(\<lambda>x. indicat_real (einterval a b) x *\<^sub>R f x) \<in> borel_measurable lborel"