hcju/isabelleps huggingface dataset:
lenarg: fixes u :: "'b list"   and v :: "'b list" assumes "u = v" shows "\<^bold>|u\<^bold>| = \<^bold>|v\<^bold>|"
index_map_inj: fixes f :: "'a \<Rightarrow> 'b"   and xs :: "'a list"   and y :: "'a" assumes "inj f" shows "index (map f xs) (f y) = index xs y"
PUF: shows "{p. p permutes UNIV} \<subseteq> {f. (\<forall>i\<in>UNIV. f i \<in> UNIV) \<and> (\<forall>i. i \<notin> UNIV \<longrightarrow> f i = i)}"
sqrt_le_D: fixes x :: "real"   and y :: "real" assumes "sqrt x \<le> y" shows "x \<le> y\<^sup>2"
leftderives_def: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and u :: "symbol list"   and v :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.leftderives \<NN> \<TT> \<RR> u v = ((u, v) \<in> CFG.leftderivations \<NN> \<TT> \<RR>)"
gen_lossless_gpv_True: assumes "co' = True" shows "local.gen_lossless_gpv \<equiv> gfp F"
card_fset: fixes xs :: "'b fset" shows "card_fset xs = card (fset xs)"
A_dim_col_not2: shows "dim_col A \<noteq> 2"
HomI: fixes \<CC> :: "V"   and a :: "V"   and b :: "V"   and f :: "V" assumes "f : a \<mapsto>\<^bsub>\<CC>\<^esub> b" shows "f \<in>\<^sub>\<circ> Hom \<CC> a b"
record_simps_5: fixes f :: "nat \<Rightarrow> nat"   and x :: "'a state" shows "defined_mods (update_clock f x) = defined_mods x"
op_set_isEmpty_def: fixes s :: "'a set" shows "op_set_isEmpty s \<equiv> s = {}"
g: shows "g \<in> M \<rightarrow>\<^sub>M count_space I"
VSLuckiest_def: shows "VSLuckiest = opt \<circ> QE_dnf opt (\<lambda>amount. luckiestFind) \<circ> opt"
except_eq_zeroI: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and S :: "'a set" assumes "keys p \<subseteq> S" shows "except p S = 0"
max_bag_card_lower_bound_bag: fixes t :: "'c" assumes "t \<in> V\<^bsub>T\<^esub>" shows "card (bag t) \<le> max_bag_card"
span_in_category_axioms: shows "span_in_category (\<cdot>) \<lparr>Leg0 = ra, Leg1 = r1\<rparr>"
rpr_less_le_trans: fixes x :: "'c"   and r :: "('c \<times> 'c) set"   and y :: "'c"   and z :: "'c"   and A :: "'c set" assumes "x \<^bsub>r\<^esub>\<prec> y"   and "y \<^bsub>r\<^esub>\<preceq> z"   and "rpr A r" shows "x \<^bsub>r\<^esub>\<prec> z"
slength_0_conv_3: fixes xs :: "[:'a:]" shows "eq\<cdot>0\<cdot>(slength\<cdot>xs) = snull\<cdot>xs"
leSup: fixes i :: "nat" shows "M i \<tturnstile> Sup_exp (range M)"
some_geodesic_is_geodesic_segment_1: fixes dist :: "'a \<Rightarrow> 'a \<Rightarrow> real"   and uniformity :: "('a \<times> 'a) filter"   and "open" :: "'a set \<Rightarrow> bool"   and x :: "'a"   and y :: "'a" assumes "class.geodesic_space dist uniformity open" shows "geodesic_segment_between {x--y} x y"
UP_cring_def: fixes R :: "('c, 'd) ring_scheme" shows "UP_cring R \<equiv> cring R"
unsatisfiableFormulaWithSingleLiteralClause: fixes formula :: "Literal list list"   and literal :: "Literal" assumes "\<not> satisfiable formula"   and "[literal] el formula" shows "formulaEntailsLiteral (removeAll [literal] formula) (opposite literal)"
is_red_singletonI: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and q :: "'t \<Rightarrow>\<^sub>0 'c" assumes "is_red F q" obtains p :: "'t \<Rightarrow>\<^sub>0 'c" where "p \<in> F" and "is_red {p} q"
2: shows "\<exists>o\<^sub>1 o\<^sub>2. Some (\<omega>\<nu> o\<^sub>1) = d\<^sub>\<kappa> x \<and> Some (\<omega>\<nu> o\<^sub>2) = d\<^sub>\<kappa> y"
assms_2: shows "\<forall>x\<in>{0..1}. \<gamma> differentiable at x"
SS_def: fixes x :: "'a" shows "SS x = {y. x - y \<in> supp f \<and> y \<in> supp g}"
in_Q0: fixes A :: "'a set" assumes "A \<in> sets M"   and "A \<inter> \<Union> (range Q) = {}" shows "emeasure M A = 0 \<and> emeasure N A = 0 \<or> 0 < emeasure M A \<and> emeasure N A = \<infinity>"
C: shows "Maps.Comp \<lbrakk>t\<^sub>0\<rbrakk> \<lbrakk>\<omega>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>1\<rbrakk> = Maps.Comp \<lbrakk>u\<^sub>1\<rbrakk> \<lbrakk>\<chi>.chine \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<rbrakk>"
code: fixes x :: "('i, 'g, 'h) Resumption.resumption"   and f :: "'i \<Rightarrow> ('f, 'g, 'h) Resumption.resumption" shows "x \<bind> f = (if Resumption.resumption.is_Done x \<and> (Resumption.resumption.result x \<noteq> None \<longrightarrow> Resumption.resumption.is_Done (f (the (Resumption.resumption.result x)))) then Resumption.resumption.Done (Resumption.resumption.result x \<bind> Resumption.resumption.result \<circ> f) else Resumption.resumption.Pause (if Resumption.resumption.is_Done x then Resumption.resumption.output (f (the (Resumption.resumption.result x))) else Resumption.resumption.output x) (\<lambda>inp. if Resumption.resumption.is_Done x then Resumption.resumption.resume (f (the (Resumption.resumption.result x))) inp else Resumption.resumption.resume x inp \<bind> f))"
len_pivot: shows "length (pivot_positions A) = card {i. i < dim_row A \<and> row A i \<noteq> 0\<^sub>v (dim_col A)}"
b1: fixes A :: "'a set set"   and z :: "'a set" assumes "Ball A closed_csubspace"   and "closed_csubspace z"   and "\<And>x. \<lbrakk>closed_csubspace x; x \<in> A\<rbrakk> \<Longrightarrow> z \<subseteq> x" shows "z \<subseteq> \<Inter> A"
span_extend_basis: fixes B :: "'b set" assumes "vs1.independent B" shows "vs1.span (vs1.extend_basis B) = UNIV"
power_polys_works: fixes i :: "nat"   and n :: "nat"   and curr_p :: "'a"   and u :: "'a"   and mult_p :: "'a" assumes "i < n"   and "curr_p = curr_p mod u" shows "power_polys mult_p u curr_p n ! i = curr_p * mult_p ^ i mod u"
extend_guarantees_imp_guarantees: fixes h :: "'d \<times> 'e \<Rightarrow> 'f"   and F :: "'d program"   and X :: "'d program set"   and Y :: "'d program set" assumes "Extend h"   and "extend h F \<in> extend h ` X guarantees extend h ` Y" shows "F \<in> X guarantees Y"
equivpE: fixes R :: "'c \<Rightarrow> 'c \<Rightarrow> bool" assumes "equivp R" obtains "reflp R" and "symp R" and "transp R"
6: shows "(w \<sqinter> - prim_EP w v e)\<^sup>\<star> * prim_P w v e\<^sup>T\<^sup>\<star> = (w \<sqinter> - prim_EP w v e)\<^sup>+ \<squnion> prim_P w v e\<^sup>T\<^sup>\<star>"
inner_prod_minus_distrib_right: fixes u :: "'c vec"   and n :: "nat"   and v :: "'c vec"   and w :: "'c vec" assumes "u \<in> carrier_vec n"   and "v \<in> carrier_vec n"   and "w \<in> carrier_vec n" shows "inner_prod u (v - w) = inner_prod u v - inner_prod u w"
shiftr_div_2n': fixes w :: "'b word"   and n :: "nat" shows "unat (w >> n) = unat w div 2 ^ n"
Diff_Int_distrib2: fixes A :: "'c set"   and B :: "'c set"   and C :: "'c set" shows "(A - B) \<inter> C = A \<inter> C - B \<inter> C"
assms: shows "xs <\<^sub>r\<^sub>l\<^sub>e\<^sub>x ys"
hidden_inacc: fixes xs :: "'a list"   and i :: "nat"   and x :: "'a" assumes "hidden xs i" shows "last_index xs x \<noteq> i"
rel_stacktraces_Nil: fixes get :: "'a \<Rightarrow> ('b, 'c, 'a, 'op, 'opinl, 'opubx, 'd, 'e) Ubx.instr fundef option"   and opt :: "'a option" shows "rel_stacktraces get [] [] opt"
apsnd_conv: fixes f :: "'e \<Rightarrow> 'd"   and x :: "'c"   and y :: "'e" shows "apsnd f (x, y) = (x, f y)"
\<R>\<^sub>1_mem_eq: fixes c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val"   and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt"   and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val"   and \<Gamma>' :: "'Var \<Rightarrow> Sec option" assumes "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>'\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" shows "mem\<^sub>1 =\<^bsub>mds\<^esub>\<^sup>l mem\<^sub>2"
set_map: fixes f :: "'b \<Rightarrow> 'a"   and v :: "'b ??.MFMC_Flow_Attainability.flow_attainability.node" shows "??.MFMC_Flow_Attainability.flow_attainability.node.set_node (??.MFMC_Flow_Attainability.flow_attainability.node.map_node f v) = f ` ??.MFMC_Flow_Attainability.flow_attainability.node.set_node v"
that: fixes p :: "'b list" assumes "trail u p v"   and "set p \<subseteq> face_cycle_set a" shows "thesis"
intros: fixes t :: "('s, 'v) tm"   and s :: "('s, 'v) tm" assumes "head t >\<^sub>h\<^sub>d head s"   and "is_Sym (head s)"   and "chkchop (>\<^sub>t) t s" shows "gt_diff t s"
fds_remainder_def: fixes m :: "nat" shows "fds_remainder m = fds_subseries ((<) m)"
minim_in: fixes r :: "('d \<times> 'd) set"   and B :: "'d set" assumes "wo_rel r"   and "B \<subseteq> Field r"   and "B \<noteq> {}" shows "minim r B \<in> B"
mult1E: fixes N :: "'a multiset"   and M :: "'a multiset"   and r :: "('a \<times> 'a) set" assumes "(N, M) \<in> mult1 r" obtains a :: "'a" and M0 :: "'a multiset" and K :: "'a multiset" where "M = add_mset a M0" and "N = M0 + K" and "\<And>b. b \<in># K \<Longrightarrow> (b, a) \<in> r"
induct: fixes P :: "'d ifex list \<Rightarrow> bool"   and a0 :: "'d ifex list" assumes "P []"   and "\<And>v uu_ uv_ r. P r \<Longrightarrow> P (IF v uu_ uv_ # r)"   and "\<And>r. P r \<Longrightarrow> P (Trueif # r)"   and "\<And>r. P r \<Longrightarrow> P (Falseif # r)" shows "P a0"
lift_StrongPostdomination: fixes sourcenode :: "'f \<Rightarrow> 'g"   and targetnode :: "'f \<Rightarrow> 'g"   and kind :: "'f \<Rightarrow> 'h edge_kind"   and valid_edge :: "'f \<Rightarrow> bool"   and Entry :: "'g"   and Def :: "'g \<Rightarrow> 'i set"   and Use :: "'g \<Rightarrow> 'i set"   and state_val :: "'h \<Rightarrow> 'i \<Rightarrow> 'j"   and Exit :: "'g"   and nx :: "'g" assumes "CFGExit_wf sourcenode targetnode kind valid_edge Entry Def Use state_val Exit"   and "StrongPostdomination sourcenode targetnode kind valid_edge Entry Exit"   and "CFGExit.inner_node sourcenode targetnode valid_edge Entry Exit nx" shows "StrongPostdomination src trg knd (lift_valid_edge valid_edge sourcenode targetnode kind Entry Exit) NewEntry NewExit"
a_set_attribute_locs_def: fixes element_ptr :: "(_) element_ptr" shows "local.a_set_attribute_locs element_ptr \<equiv> all_args (put_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t element_ptr attrs_update)"
dim_hyp1: shows "M_mat signs subsets \<in> carrier_mat (length signs) (length signs)"
gen2_ne: fixes m :: "nat"   and n :: "nat"   and b :: "nat list"   and c :: "nat list" shows "Algorithm.gen2 m n b c \<noteq> []"
lens_put_comm: fixes X :: "'b \<Longrightarrow> 'c"   and Y :: "'d \<Longrightarrow> 'c"   and \<sigma> :: "'c"   and v :: "'d"   and u :: "'b" assumes "X \<bowtie> Y" shows "put\<^bsub>X\<^esub> (put\<^bsub>Y\<^esub> \<sigma> v) u = put\<^bsub>Y\<^esub> (put\<^bsub>X\<^esub> \<sigma> u) v"
empty_replicate: fixes n :: "nat"   and x :: "'g" shows "([] = replicate n x) = (n = 0)"
inv_Ex2: fixes f :: "nat \<Rightarrow> nat" assumes "f \<in> Ex2" shows "inv f \<in> Ex2"
atLeastLessThan_empty_iff2: fixes a :: "'d multiset"   and b :: "'d multiset" shows "({} = subset_mset.atLeastLessThan a b) = (\<not> a \<subset># b)"
EmptyMap_dom: fixes S :: "'d set" shows "dom (EmptyMap S) = S"
normal_oVeblen: fixes x :: "ordinal" shows "normal (oVeblen F x)"
runit'_simps_2: fixes f :: "'a" assumes "S.ide f" shows "S.src (S.runit' f) = S.src f"
choice: shows "rel = Lt_Rel \<and> d \<le> 0 \<or> rel = Leq_Rel \<and> d < 0"
det_zero_low_rank: fixes A :: "'a mat"   and n :: "nat" assumes "A \<in> carrier_mat n n"   and "det A = (0::'a)" shows "vec_space.rank n A < n"
npowf_mem: fixes a :: "'a"   and n :: "int" assumes "a \<in> carrier K"   and "a \<noteq> \<zero>" shows "a\<^bsub>K\<^esub>\<^bsup>n\<^esup> \<in> carrier K"
HStartBallot_HInv4a1: fixes s :: "state"   and s' :: "state"   and p :: "Proc" assumes "HStartBallot s s' p"   and "HInv4a1 s p"   and "Inv2a_inner s' p" shows "HInv4a1 s' p"
d_true: shows "I \<Turnstile> DA \<cdot> \<sigma> \<cdot> \<eta>"
fp_cop_F_allocation: fixes ds :: "'b set" shows "inj_on Xd (cop ds)"
on_triple_def: fixes R :: "('c \<times> 'd) set" shows "on_triple R \<equiv> {((l, s, t), l', s', t'). l = l' \<and> (s, s') \<in> R \<and> (t, t') \<in> R}"
differentiable_imp_continuous_on: fixes f :: "'c \<Rightarrow> 'd"   and s :: "'c set" assumes "f differentiable_on s" shows "continuous_on s f"
contract\<Gamma>\<Gamma>: fixes \<Gamma> :: "'b formula multiset"   and \<Gamma>' :: "'b formula multiset"   and \<Delta> :: "'b formula multiset" assumes "\<Gamma> + \<Gamma>' \<Rightarrow> \<Delta>"   and "\<Gamma>' \<subseteq># \<Gamma>" shows "\<Gamma> \<Rightarrow> \<Delta>"
card_partition_on: fixes A :: "'b set"   and k :: "nat" assumes "finite A" shows "card {P. partition_on A P \<and> card P = k} = Stirling (card A) k"
fetch_instr_result_1: fixes s' :: "'b sparc_state" assumes "\<nexists>e. fetch_instruction s' = Inl e" shows "\<exists>v. fetch_instruction s' = Inr v"
irreducible_root_free: fixes p :: "'a poly" assumes "irreducible p" shows "root_free p"
resumption_ord_is_DoneD: fixes r :: "('d, 'e, 'f) Resumption.resumption"   and r' :: "('d, 'e, 'f) Resumption.resumption" assumes "resumption_ord r r'"   and "Resumption.resumption.is_Done r'" shows "Resumption.resumption.is_Done r"
two_pred_two_add_once: fixes c :: "one\<^sub>\<bottom>" shows "c \<sqsubseteq> two_pred\<cdot>(two_add\<cdot>once\<cdot>c)"
\<section>: shows "{b. \<exists>T. openin (top_of_set S) T \<and> b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> Q x)} \<inter> {b. \<exists>T. openin (top_of_set S) T \<and> b \<in> T \<and> (\<forall>x\<in>T. P x \<longrightarrow> \<not> Q x)} = {}"
B_def: shows "B = 1\<^sub>m nr @\<^sub>c A @\<^sub>c - A"
assms_4: shows "Nom k at j in (ps, i) # branch"
sumTT'_2: shows "sum c' T' = 1 - c' a"
3: fixes t :: "(('fun, 'atom, 'sets) prot_fun, (('fun, 'atom, 'sets) prot_fun, 'atom prot_atom) Term.term \<times> nat) Term.term" assumes "t \<in> set M"   and "wf\<^sub>t\<^sub>r\<^sub>m t" shows "comp_timpl_closure {t} (set TI') \<subseteq> timpl_closure_set (set M) (set TI)"
bound_14_OT: fixes a1 :: "bool"   and a2 :: "bool"   and A :: "'v_14_OT1 \<Rightarrow> bool \<Rightarrow> bool spmf"   and b1 :: "bool"   and b2 :: "bool" shows "\<bar>spmf (coin_spmf \<bind> (\<lambda>\<sigma>. S1_14_OT (\<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False), \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True), \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False), \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True)) () \<bind> (\<lambda>view. A view \<sigma>))) True - spmf (coin_spmf \<bind> (\<lambda>\<sigma>. R1_14_OT (\<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> False), \<sigma> \<oplus> (a1 \<oplus> False \<and> a2 \<oplus> True), \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> False), \<sigma> \<oplus> (a1 \<oplus> True \<and> a2 \<oplus> True)) (b1, b2) \<bind> (\<lambda>view. A view \<sigma>))) True\<bar> \<le> adv_14_OT"
has_bochner_integral_eq: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "has_bochner_integral M f x"   and "has_bochner_integral M f y" shows "x = y"
the_elem_wf': fixes X :: "'a set" assumes "X \<in> carrier (R Quot a_kernel R S h)" shows "\<exists>r\<in>carrier R. h ` X = {h r}"
oldest_token_distinct: fixes \<Sigma> :: "'c set"   and \<delta> :: "'d \<Rightarrow> 'c \<Rightarrow> 'd"   and q\<^sub>0 :: "'d"   and w :: "nat \<Rightarrow> 'c"   and q :: "'d"   and q' :: "'d"   and n :: "nat"   and i :: "nat"   and j :: "nat" assumes "semi_mojmir \<Sigma> \<delta> q\<^sub>0 w"   and "q \<noteq> q'"   and "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w q n = Some i"   and "semi_mojmir_def.oldest_token \<delta> q\<^sub>0 w q' n = Some j" shows "i \<noteq> j"
Fvars_fmla_simps_4: fixes A :: "fmla"   and B :: "fmla" shows "Fvars (A IMP B) = Fvars A \<union> Fvars B"
shift_mmuaux_eq: shows "shift_mmuaux args nt aux = foldl (\<lambda>aux _. eval_step_mmuaux aux) aux tss_list"
inference_closed_sets_are_saturated: fixes S :: "'a eclause set" assumes "inference_closed S"   and "\<forall>x\<in>S. finite (cl_ecl x)" shows "clause_saturated S"
assms: shows "\<forall>l\<in>dom f. \<forall>x y. Px x y \<longrightarrow> P x y (f l)"
gx0_le_gx1: shows "gx0 \<le> gx1"
subst_term_fv: fixes x :: "variable"   and \<tau> :: "typ"   and y :: "variable"   and t :: "Core.term" shows "fv (subst_term [((x, \<tau>), Fv y \<tau>)] t) = (if (x, \<tau>) \<in> fv t then insert (y, \<tau>) else id) (fv t - {(x, \<tau>)})"
Arity_up_zero_join2: fixes x :: "Arity\<^sub>\<bottom>" shows "up\<cdot>0 \<squnion> x = up\<cdot>0"
BFs_1: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<^bold>\<box>\<phi> \<alpha>) \<^bold>\<rightarrow> \<^bold>\<box>(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) in v]"
mset_ran_empty_iff: fixes r :: "int set"   and a :: "int \<Rightarrow> 'a" assumes "finite r" shows "(mset_ran a r = {#}) = (r = {})"
added_ids_Deliver_Expunge_diff_collapse: fixes e :: "'b"   and e' :: "'b"   and i :: "'c"   and mo :: "'c"   and j :: "'c" assumes "e \<noteq> e'" shows "added_ids [Deliver (i, Expunge e mo j)] e' = []"
perp_col2_bis: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and D :: "'p"   and P :: "'p"   and Q :: "'p" assumes "A B Perp C D"   and "Col C D P"   and "Col C D Q"   and "P \<noteq> Q" shows "A B Perp P Q"
lub_bool: fixes S :: "bool set" shows "lub S = (True \<in> S)"
last_token_of_path: fixes q :: "(symbol \<times> character list) list"   and k :: "nat"   and u :: "nat"   and q' :: "(symbol \<times> character list) list"   and t :: "symbol \<times> character list" assumes "q \<in> \<P> k u"   and "q = q' @ [t]"   and "charslength q' = k" shows "t \<in> \<Z> k u"
subst_term_not_loose_bvar: fixes t :: "Core.term"   and n :: "nat"   and b :: "Core.term"   and idn :: "variable"   and T :: "typ" assumes "\<not> loose_bvar t n"   and "is_closed b" shows "\<not> loose_bvar (subst_term [((idn, T), b)] t) n"
axioms_1: fixes \<alpha> :: "V"   and \<BB> :: "V"   and \<CC> :: "V" assumes "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^sub>.\<^sub>w\<^sub>i\<^sub>d\<^sub>e\<^bsub>\<alpha>\<^esub> \<CC>" shows "\<BB> \<subseteq>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<CC>"
ord'_add0: fixes k1 :: "int option"   and t :: "'a tree23"   and k2 :: "int option"   and k :: "int"   and y :: "'a" assumes "ord' k1 t k2"   and "opt_less k1 (Some k)"   and "opt_less (Some k) k2" shows "ord' k1 (add0 k y t) k2"
collapse: fixes state :: "'b state" shows "make_state (clock state) (refs state) (ffi state) (defined_types state) (defined_mods state) = state"
i_set_imp_not_empty: fixes I :: "nat set" assumes "I \<in> i_set" shows "I \<noteq> {}"
assms_13: fixes ws :: "int poly multiset" assumes "ws \<subseteq># mset vs"   and "ws \<noteq> {#}"   and "size ws < d \<or> size ws = d \<and> ws \<notin> (mset \<circ> snd) ` set cands" shows "test_dvd u ws"
cong_trans: fixes b :: "'b"   and c :: "'b"   and a :: "'b"   and d :: "'b" assumes "[b = c] (mod a)"   and "[c = d] (mod a)" shows "[b = d] (mod a)"
matches_simps_2: fixes \<gamma> :: "('a \<Rightarrow> 'b \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'b \<Rightarrow> bool)"   and m :: "'a match_expr"   and a :: "action"   and p :: "'b"   and m' :: "'a match_expr" assumes "matches \<gamma> m a p" shows "matches \<gamma> (MatchAnd m' m) a p = matches \<gamma> m' a p"
irrefl_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "RPRs.irrefl A r \<equiv> r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. (x, x) \<notin> r)"
power_Suc2: fixes one :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b"   and n :: "nat" assumes "class.monoid_mult one times" shows "power.power one times a (Suc n) = times (power.power one times a n) a"
norm_uminus_minus: fixes minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and dist :: "'b \<Rightarrow> 'b \<Rightarrow> real"   and norm :: "'b \<Rightarrow> real"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and uminus :: "'b \<Rightarrow> 'b"   and scaleR :: "real \<Rightarrow> 'b \<Rightarrow> 'b"   and sgn :: "'b \<Rightarrow> 'b"   and uniformity :: "('b \<times> 'b) filter"   and "open" :: "'b set \<Rightarrow> bool"   and x :: "'b"   and y :: "'b" assumes "class.real_normed_vector minus dist norm plus zero uminus scaleR sgn uniformity open" shows "norm (minus (uminus x) y) = norm (plus x y)"
uint_up_ucast: fixes w :: "'b word" assumes "is_up ucast" shows "uint (ucast w) = uint w"
openin_trans: fixes T :: "'b set"   and S :: "'b set"   and U :: "'b set" assumes "openin (top_of_set T) S"   and "openin (top_of_set U) T" shows "openin (top_of_set U) S"
jj: shows "j \<le> (GREATEST j. k cd\<^bsup>\<pi>\<^esup>\<rightarrow> j \<and> ipd (\<pi> j) \<noteq> \<pi> m)"
adjust_div_eq: fixes q :: "int"   and r :: "int" shows "Divides.adjust_div (q, r) = q + of_bool (r \<noteq> 0)"
simps_1: fixes ys :: "'l list" shows "[] @ ys = ys"
merge_state_self: fixes s :: "nat"   and t :: "(nat list \<times> (nat \<times> nat) \<times> transition) fset" shows "merge_states s s t = t"
atLeastPlusOneLessThan_greaterThanLessThan_int: fixes l :: "int"   and u :: "int" shows "{l + 1..<u} = {l<..<u}"
span_subset_spanI: fixes A :: "('t \<Rightarrow>\<^sub>0 'c) set"   and B :: "('t \<Rightarrow>\<^sub>0 'c) set" assumes "A \<subseteq> pmdl B" shows "pmdl A \<subseteq> pmdl B"
appliedDecide_def: fixes stateA :: "State"   and stateB :: "State"   and decisionVars :: "nat set" shows "appliedDecide stateA stateB decisionVars \<equiv> \<exists>l. var l \<in> decisionVars \<and> \<not> l el elements (getM stateA) \<and> \<not> opposite l el elements (getM stateA) \<and> getM stateB = getM stateA @ [(l, True)]"
cases: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" obtains "x \<sqsubset> y"   | "x = y"   | "y \<sqsubset> x"
set_map: fixes f :: "'f \<Rightarrow> 'e"   and v :: "'f option" shows "set_option (map_option f v) = f ` set_option v"
bypass: fixes z :: "'a" assumes "x \<noteq> y"   and "z \<in> set p" shows "z = x \<or> z \<notin> S"
Complex_eq_numeral: fixes a :: "real"   and b :: "real"   and w :: "num" shows "(Complex a b = numeral w) = (a = numeral w \<and> b = 0)"
mult_diff_mult: fixes x :: "'c"   and y :: "'c"   and a :: "'c"   and b :: "'c" shows "x * y - a * b = x * (y - b) + (x - a) * b"
eventually_at_top_linorder: fixes P :: "'e \<Rightarrow> bool" shows "eventually P at_top = (\<exists>N. \<forall>n\<ge>N. P n)"
rel_fun_mono: fixes X :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and f :: "'d \<Rightarrow> 'f"   and g :: "'e \<Rightarrow> 'g"   and Y :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and B :: "'f \<Rightarrow> 'g \<Rightarrow> bool" assumes "rel_fun X A f g"   and "\<And>x y. Y x y \<longrightarrow> X x y"   and "\<And>x y. A x y \<Longrightarrow> B x y" shows "rel_fun Y B f g"
union_vars_monom_alt_def: fixes xs :: "char list list"   and \<V> :: "char list set" shows "union_vars_monom xs \<V> = \<V> \<union> set xs"
ctxt_ctxt_compose: fixes C :: "('c, 'd) ctxt"   and D :: "('c, 'd) ctxt"   and t :: "('c, 'd) Term.term" shows "(C \<circ>\<^sub>c D)\<langle>t\<rangle> = C\<langle>D\<langle>t\<rangle>\<rangle>"
mk_poly_sylvester_upper: fixes i :: "nat"   and q :: "'b poly"   and p :: "'b poly" assumes "i < degree q" shows "mk_poly (sylvester_mat p q) $$ (i, degree p + degree q - 1) = monom (1::'b) (degree q - Suc i) * p"
seq_stop: fixes c :: "('Var, 'AExp, 'BExp) Stmt"   and mem :: "'Var \<Rightarrow> 'Val" shows "eval\<^sub>w_simplep (Stop ;; c) mem c mem"
not_final_thread_cases2: fixes s :: "('a, 'b) locks \<times> (('b \<Rightarrow> ('x \<times> 'a \<Rightarrow>f nat) option) \<times> 'c) \<times> ('b \<Rightarrow> 'd wait_set_status option) \<times> 'b set"   and t :: "'b"   and x :: "'x"   and ln :: "'a \<Rightarrow>f nat" assumes "not_final_thread s t"   and "thr s t = \<lfloor>(x, ln)\<rfloor>" obtains "\<not> final x"   | "ln \<noteq> no_wait_locks"   | w :: "'d wait_set_status" where "wset s t = \<lfloor>w\<rfloor>"
wf_listn': shows "wf {(y, x). x [\<sqsubset>\<^bsub>r\<^esub>] y}"
prod_5: fixes a :: "'a N"   and b :: "'a N"   and c :: "'a N"   and f :: "'a N" shows "((a l\<rightarrow> b) l\<rightarrow> b) * (c * a l\<rightarrow> f * a) * (c * b l\<rightarrow> f * b) \<le> c l\<rightarrow> f"
zeroU_stateT_def: shows "zeroU = StateT\<cdot>(\<Lambda> s. mzero)"
in_qtableE: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and x :: "'a option list" assumes "qtable n A P Q X"   and "x \<in> X"   and "P x" obtains "wf_tuple n A x" and "Q x"
Ide_Cod: fixes t :: "'a local.term" assumes "Arr t" shows "Ide (Cod t)"