hcju/isabelleps huggingface dataset:
CatExpMorNT: fixes f :: "('d, 'e, 'f, 'g, 'h, 'i) NatTrans"   and A :: "('d, 'f, 'h) Category_scheme"   and B :: "('e, 'g, 'i) Category_scheme" assumes "f \<in> mor\<^bsub>CatExp A B\<^esub>" shows "NatTrans f"
pickInp_def: fixes inp :: "'c \<Rightarrow> 'd set option" shows "pickInp inp \<equiv> lift pick inp"
nucleus_alt_def: fixes f :: "'b \<Rightarrow> 'b" shows "nucleus f = (\<forall>x y. f x \<rightarrow> f y = x \<rightarrow> f y \<and> f y \<leftarrow> f x = f y \<leftarrow> x)"
a1: shows "H (\<tt>\<tt> p \<cdot> \<tt>\<tt> r) x p"
enc_unique_length: fixes v :: "('a \<times> bool list) list"   and w :: "'a list"   and I :: "(nat + nat set) list" assumes "v \<in> enc (w, I)" shows "\<forall>v'. length v' = length v \<and> v' \<in> enc (w, I) \<longrightarrow> v = v'"
S1: shows "c_assoc_have_key ls (c_pair (c_fst (c_snd (c_fst key))) (c_snd key)) = 0"
Says_certificate_valid: fixes A :: "agent"   and B :: "agent"   and SK :: "nat"   and lid :: "msg"   and xid :: "msg"   and cc :: "msg"   and cm :: "msg"   and C :: "agent"   and EK :: "nat"   and evs :: "event list" assumes "Says A B (sign SK \<lbrace>lid, xid, cc, cm, cert C EK onlyEnc (priSK RCA)\<rbrace>) \<in> set evs"   and "evs \<in> set_pur" shows "EK = pubEK C"
updEnv_def: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('d, 'e, 'varSort, 'var, 'f) qTerm set option"   and x :: "'var"   and X :: "('d, 'e, 'varSort, 'var, 'f) qTerm set"   and xs :: "'varSort" shows "rho [x \<leftarrow> X]_xs \<equiv> \<lambda>ys y. if ys = xs \<and> y = x then Some X else rho ys y"
encode_problem_parallel_correct_x: fixes \<Pi> :: "'b strips_problem"   and \<A> :: "sat_plan_variable \<Rightarrow> bool"   and t :: "nat" assumes "is_valid_problem_strips \<Pi>"   and "\<A> \<Turnstile> \<Phi> \<Pi> t" shows "map (decode_state_at \<Pi> \<A>) [0..<Suc (length (\<Phi>\<inverse> \<Pi> \<A> t))] = trace_parallel_plan_strips (\<Pi>\<^sub>I) (\<Phi>\<inverse> \<Pi> \<A> t)"
intensional_rews_2: fixes f :: "'d \<Rightarrow> 'c"   and x :: "'e \<Rightarrow> 'd"   and w :: "'e" shows "(f<x>) w \<equiv> f (x w)"
upto_simps_2: fixes m :: "int"   and n :: "int" assumes "m \<le> n" shows "upto\<cdot>(MkI\<cdot>m)\<cdot>(MkI\<cdot>n) = MkI\<cdot>m : [MkI\<cdot>m + 1..MkI\<cdot>n]"
intros_1: fixes n :: "C"   and b :: "nat" assumes "n \<in> hbase b" shows "n \<in> hbase_ext b"
that: assumes "T = r ` S"   and "r ` S \<subseteq> S"   and "continuous_on S r"   and "\<And>x. x \<in> S \<Longrightarrow> r (r x) = r x" shows "thesis"
neighborhoods_def: fixes x :: "'a set" shows "target.neighborhoods x \<equiv> {U. is_zariski_open U \<and> x \<in> U}"
single_one: shows "monomial (1::'b) (0::'a) = 1"
fresh_inner_floatariths: fixes xs :: "floatarith list"   and ys :: "floatarith list"   and i :: "nat" assumes "length xs = length ys" shows "fresh_floatarith (inner_floatariths xs ys) i = (fresh_floatariths xs i \<and> fresh_floatariths ys i)"
op_image_fst_ivl_coll_def: fixes X :: "('c \<times> 'd) set" shows "op_image_fst_ivl_coll X = sets_of_coll X \<bind> (\<lambda>Xs. FORWEAK Xs (RETURN op_empty_coll) (\<lambda>X. op_image_fst_ivl X \<bind> (\<lambda>i. RETURN (mk_coll i))) (\<lambda>X' X. RETURN (X' \<union> X)))"
ode_na_def: shows "ode_na = (\<lambda>a. ode (snd a))"
subdiv_step: shows "subdivision_step G rev_G H rev_H (u, v, w) (uv, uw, vw)"
inject_5: fixes x51 :: "nat"   and x52 :: "'d"   and x53 :: "'d"   and y51 :: "nat"   and y52 :: "'d"   and y53 :: "'d" shows "(RecvMarker x51 x52 x53 = RecvMarker y51 y52 y53) = (x51 = y51 \<and> x52 = y52 \<and> x53 = y53)"
le_neq_trans: fixes a :: "complex"   and b :: "complex" assumes "a \<le>\<^sub>\<complex> b"   and "a \<noteq> b" shows "a <\<^sub>\<complex> b"
matrix_to_iarray_scalar_matrix_mult: fixes k :: "'b"   and A :: "(('b, 'c) vec, 'd) vec" shows "matrix_to_iarray (k *k A) = k *ssi matrix_to_iarray A"
I2: fixes S :: "'b"   and x :: "'a"   and xs :: "'a list" assumes "invariant S"   and "isNode x"   and "list_all isNode xs"   and "memb x S \<Longrightarrow> P S xs"   and "\<not> memb x S \<Longrightarrow> P (ins x S) (succs x @ xs)" shows "P S (x # xs)"
sgn1_def: fixes x :: "nat" shows "sgn1 x = (case x of 0 \<Rightarrow> 0 | Suc y \<Rightarrow> 1)"
preserves_seq: fixes g :: "'b"   and f :: "'b" assumes "B.seq g f" shows "C.seq (F g) (F f)"
min_key_val_raw_alt: fixes xs :: "('a \<times> 'c) list \<times> 'o"   and ko :: "'o" assumes "oalist_inv xs"   and "fst xs \<noteq> []" shows "min_key_val_raw ko xs = list.hd (sort_oalist_aux ko xs)"
assms_2: shows "(a, b) \<in> r\<^sup>*"
comp_assoc: fixes f :: "'j \<Rightarrow> 'h"   and g :: "'i \<Rightarrow> 'j"   and h :: "'g \<Rightarrow> 'i" shows "f \<circ> g \<circ> h = f \<circ> (g \<circ> h)"
INF_eq_const: fixes Inf :: "'a set \<Rightarrow> 'a"   and Sup :: "'a set \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and I :: "'b set"   and f :: "'b \<Rightarrow> 'a"   and x :: "'a" assumes "class.complete_lattice Inf Sup inf less_eq less sup bot top"   and "I \<noteq> {}"   and "\<And>i. i \<in> I \<Longrightarrow> f i = x" shows "Inf (f ` I) = x"
ifex_vars_subset: fixes x :: "'a"   and i :: "'a ifex"   and t :: "'a ifex"   and e :: "'a ifex" assumes "x \<in> ifex_var_set (ifex_ite i t e)" shows "x \<in> ifex_var_set i \<or> x \<in> ifex_var_set t \<or> x \<in> ifex_var_set e"
allDefs_def: fixes g :: "'g"   and n :: "'node" shows "braun_ssa.allDefs g n \<equiv> defs' g n \<union> braun_ssa.phiDefs g n"
simps_2: fixes s :: "('c, 'd) fmap"   and a :: "('c, 'd) fmap \<times> ('c, 'd) fmap"   and as :: "(('c, 'd) fmap \<times> ('c, 'd) fmap) list" shows "sat_precond_as s (a # as) = (fst a \<subseteq>\<^sub>f s \<and> sat_precond_as (state_succ s a) as)"
intros_1: fixes \<pi> :: "Syntax.term list"   and e :: "Syntax.term"   and uu :: "mode" shows "\<lless>\<pi>, e\<ggreater> uu\<rightarrow>0 \<lless>\<pi>, e\<ggreater>"
flip: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd" assumes "bounded_cbilinear prod" shows "bounded_cbilinear (\<lambda>x y. prod y x)"
prop_prop_nec_1: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[\<^bold>\<diamond>(\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p)) \<^bold>\<rightarrow> (\<^bold>\<exists>p. F \<^bold>= (\<^bold>\<lambda>x. p)) in v]"
mono: fixes xs :: "'a llist" shows "mono_llist (\<lambda>ltakeWhile. F ltakeWhile xs)"
sameDom_lift2_simp1: fixes inp1' :: "'a \<Rightarrow> 'b option"   and inp1 :: "'a \<Rightarrow> 'c option"   and inp :: "'a \<Rightarrow> 'd option"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> 'e" assumes "sameDom inp1' inp1" shows "sameDom inp (lift2 f inp1' inp1) = sameDom inp inp1'"
fmdom'_notD: fixes x :: "'b"   and m :: "('b, 'c) fmap" assumes "x \<notin> fmdom' m" shows "fmlookup m x = None"
5: shows "\<Union> (transition A a ` P) \<subseteq> nodes A"
minus_interval_def: fixes a :: "'a interval"   and b :: "'a interval" shows "a - b = a + - b"
solve_equation: fixes H :: "int set"   and x :: "int"   and y :: "int" assumes "subgroup H (Z n)"   and "x \<in> H"   and "y \<in> H" shows "\<exists>h\<in>H. y = h \<otimes>\<^bsub>Z n\<^esub> x"
inj_on_arev: fixes G :: "('c, 'd) pre_digraph"   and arev :: "'d \<Rightarrow> 'd"   and S :: "'d set" assumes "bidirected_digraph G arev" shows "inj_on arev S"
assms: shows "[\<^bold>\<box>(\<^bold>\<forall>F. \<phi> F \<^bold>\<rightarrow> \<^bold>\<box>\<phi> F) in v]"
add_minus_cancel: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a" assumes "class.group_add minus plus zero uminus" shows "plus a (plus (uminus a) b) = b"
assms_3: fixes \<sigma> :: "'a" shows "Q' \<sigma> \<longrightarrow> Q \<sigma>"
\<epsilon>_def: fixes T :: "Unary_PCF.type"   and i :: "nat" shows "\<epsilon> T i = \<pi> T i 0 \<rightarrow> map (\<pi> T i \<circ> Suc) [0..<ar (T !- i)] \<rightarrow>\<rightarrow> T"
nres_of_simps_1: shows "nres_of dSUCCEED = SUCCEED"
pre_le_one: shows "pre \<le>\<^sub>L 1\<^sub>m N"
is_processT3_ST_pref: fixes t :: "'b event list"   and P :: "'b process"   and s :: "'b event list" assumes "t \<in> T P"   and "s \<le> t" shows "s \<in> T P"
clock_numbering_2: shows "\<forall>k\<le>n. 0 < k \<longrightarrow> (\<exists>c\<in>X. v c = k)"
simps_4: fixes f1 :: "int \<Rightarrow> 'b"   and f2 :: "'b"   and x1 :: "int" shows "(case Const x1 of Const x \<Rightarrow> f1 x | Any \<Rightarrow> f2) = f1 x1"
closed_under_renaming_def: fixes S :: "'b eclause set" shows "closed_under_renaming S = (\<forall>C D. C \<in> S \<longrightarrow> renaming_cl C D \<longrightarrow> D \<in> S)"
simps_3: fixes typeof_addr :: "'b \<Rightarrow> 'c \<Rightarrow> htype option"   and h :: "'b" shows "heap_base.typeof_h typeof_addr h Null = typeof Null"
isoI: fixes f :: "'a \<times> 'a"   and g :: "'a \<times> 'a" assumes "B.VV.inverse_arrows f g" shows "B.VV.iso f"
rev_replicate: fixes n :: "nat"   and x :: "'a" shows "rev (x \<up> n) = x \<up> n"
freshBottoChain_1: fixes xvec :: "name list" shows "xvec \<sharp>* \<one>"
greatest_not_card: shows "Suc (mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A)) \<noteq> nrows A"
abc_abd_bcdbdc: fixes a :: "'a"   and b :: "'a"   and c :: "'a"   and d :: "'a" assumes "[[a b c]]"   and "[[a b d]]"   and "c \<noteq> d" shows "[[b c d]] \<or> [[b d c]]"
initial_invar: shows "wa_initial (det_wa_wa DWA) \<subseteq> wa_invar (det_wa_wa DWA)"
graph_def: fixes A :: "('b, 'a) nba"   and w :: "'b stream"   and k :: "nat" shows "graph A w k \<equiv> alternate (clean A w) (prune A w) k (gunodes A w)"
select_convs_1: fixes nodes :: "'c set"   and edges :: "('c \<times> 'd \<times> 'c) set"   and more :: "'e" shows "nodes \<lparr>nodes = nodes, edges = edges, \<dots> = more\<rparr> = nodes"
nth_rotate: fixes ls :: "'a list"   and n :: "nat"   and m :: "nat" assumes "0 < |ls|" shows "ls ! ((n + m) mod |ls|) = rotate m ls ! (n mod |ls|)"
map_consI_2: fixes w :: "'b list"   and l :: "'b list"   and f :: "'c \<Rightarrow> 'b"   and ww :: "'c list"   and a :: "'c" assumes "w @ l = map f ww @ l" shows "f a # w @ l = map f (a # ww) @ l"
awalk_verts_arc1: fixes e :: "'b"   and p :: "'b list"   and u :: "'a" assumes "e \<in> set p" shows "tail (G \<restriction> S) e \<in> set (H.awalk_verts u p)"
map2_acom_def: fixes f :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and C1 :: "'a acom"   and C2 :: "'a acom" shows "map2_acom f C1 C2 = annotate (\<lambda>p. f (anno C1 p) (anno C2 p)) (strip C1)"
adjacent_edges_not_on_path: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat"   and v :: "nat" assumes "isPath s p t"   and "v \<notin> set (pathVertices_fwd s p)" shows "adjacent v \<inter> set p = {}"
replicate_eq_replicate: fixes m :: "nat"   and x :: "'b"   and n :: "nat"   and y :: "'b" shows "(replicate m x = replicate n y) = (m = n \<and> (m \<noteq> 0 \<longrightarrow> x = y))"
simps_5: fixes v :: "'a formula"   and va :: "'a formula" shows "is_mini_formula (v \<^bold>\<and> va) = False"
get_state_def: fixes f :: "'s \<Rightarrow> ('s, 'm) stateT" shows "local.get_state f = StateT (\<lambda>s. run_state (f s) s)"
distinct_187: fixes x71 :: "Syntax.term"   and x72 :: "Syntax.term"   and x14 :: "Syntax.term" shows "Syntax.Pair x71 x72 \<noteq> Unroll x14"
mem_ordinal_number_Worder1: fixes x :: "'c Order set"   and X :: "'c Order" assumes "x \<in> ODnums"   and "X \<in> x" shows "Worder X"
yun_factorization_def: fixes Gcd :: "'b poly \<Rightarrow> 'b poly \<Rightarrow> 'b poly"   and p :: "'b poly" shows "yun_factorization Gcd p = (if p = 0 then (0::'b, []) else let c = lead_coeff p; q = smult (inverse c) p in (c, yun_gcd.yun_monic_factorization Gcd q))"
assms_2: shows "hn_refine \<Gamma>' c \<Gamma>'' R a"
A: shows "g [^]\<^bsub>UP R\<^esub> n = g [^]\<^bsub>UP (R\<lparr>carrier := S\<rparr>)\<^esub> n"
1: fixes \<mu> :: "'a arrow_of_spans_data" assumes "V.arr \<mu>" shows "V.arr (trg \<mu>)"
quot_rep: fixes A :: "'b quot" shows "\<exists>a. A = \<lfloor>a\<rfloor>"
weak_distributivity2: fixes A1 :: "int poly fract list list"   and B1 :: "int poly fract list list"   and B2 :: "int poly fract list list" assumes "weak_matrix_match2 A1 B1 B2" shows "A1 \<otimes> rat_poly.matrix_mult B1 B2 = rat_poly.matrix_mult (A1 \<otimes> B1) B2"
refined_partition_finite: fixes R1 :: "('a \<times> 'a) set"   and R2 :: "('a \<times> 'a) set" assumes "finite (UNIV // R1)"   and "R1 \<subseteq> R2"   and "equiv UNIV R1"   and "equiv UNIV R2" shows "finite (UNIV // R2)"
csegment_midpoint_subset: fixes a :: "'a"   and b :: "'a" shows "closed_segment (midpoint a b) b \<subseteq> closed_segment a b"
member_of_subclseq_declC: fixes G :: "prog"   and m :: "qtname \<times> memberdecl"   and C :: "qtname" assumes "G \<turnstile> m member_of C" shows "G\<turnstile>C\<preceq>\<^sub>C declclass m"
pre_str_hoare_r: fixes p\<^sub>1 :: "'a upred"   and p\<^sub>2 :: "'a upred"   and C :: "'a hrel"   and q :: "'a upred" assumes "`p\<^sub>1 \<Rightarrow> p\<^sub>2`"   and "\<lbrace>p\<^sub>2\<rbrace> C \<lbrace>q\<rbrace>\<^sub>u" shows "\<lbrace>p\<^sub>1\<rbrace> C \<lbrace>q\<rbrace>\<^sub>u"
ground_action_path_in_paper_1: fixes M :: "object atom formula set"   and M' :: "object atom formula set" shows "ground_action_path M [] M' = (M = M')"
simps_2: fixes \<theta> :: "nat \<Rightarrow> nat option"   and p :: "'a"   and r :: "'a memref"   and rs :: "'a memref list" shows "history_consistent \<theta> p (r # rs) = (case r of Write\<^sub>s\<^sub>b vol a (D, f) v x xa xb xc \<Rightarrow> D \<subseteq> dom \<theta> \<and> f \<theta> = v \<and> D \<inter> read_tmps rs = {} \<and> history_consistent \<theta> p rs | Read\<^sub>s\<^sub>b vol a t v \<Rightarrow> case \<theta> t of None \<Rightarrow> False | Some v' \<Rightarrow> v = v' \<and> history_consistent \<theta> p rs | Prog\<^sub>s\<^sub>b p\<^sub>1 p\<^sub>2 is \<Rightarrow> p\<^sub>1 = p \<and> \<theta> |` (dom \<theta> - read_tmps rs)\<turnstile> p\<^sub>1 \<rightarrow>\<^sub>p (p\<^sub>2, is) \<and> history_consistent \<theta> p\<^sub>2 rs | Ghost\<^sub>s\<^sub>b set1 set2 set3 set4 \<Rightarrow> history_consistent \<theta> p rs)"
PPs_def: fixes X :: "'a set" shows ".[X] = {t. keys t \<subseteq> X}"
ftendsw: shows "((\<lambda>n. f (z n)) \<circ> K) \<longlonglongrightarrow> w"
hyps_5: assumes "xs = [] \<or> comp k (fst (last xs)) = Gt" shows "update_by_fun_pair k f xs = xs @ (if f (0::'b) = (0::'b) then [] else [(k, f (0::'b))])"
cyclic_groupI2: fixes a :: "'c" assumes "a \<in> carrier H" shows "Finite_And_Cyclic_Groups.cyclic_group (H\<lparr>carrier := generate H {a}\<rparr>) a"
discharge_invar_init: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and s :: "nat"   and t :: "nat"   and f :: "nat \<times> nat \<Rightarrow> 'a"   and l :: "nat \<Rightarrow> nat"   and n :: "nat \<Rightarrow> nat set"   and u :: "nat" assumes "neighbor_invar c s t f l n"   and "u \<in> Graph.V c - {s, t}" shows "discharge_invar c s t u f l n f l n"
equiv'_def: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and sim :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and PA :: "'c \<Rightarrow> bool"   and PB :: "'d \<Rightarrow> bool" assumes "Bisimulation_Invariant A B sim PA PB" shows "Bisimulation_Invariant.equiv' sim PA PB \<equiv> \<lambda>a b. sim a b \<and> PA a \<and> PB b"
invfun_mem1: fixes f :: "'e \<Rightarrow> 'f"   and A :: "'e set"   and B :: "'f set"   and b :: "'f" assumes "f \<in> A \<rightarrow> B"   and "bij_to f A B"   and "b \<in> B" shows "(f\<inverse>\<^bsub>B,A\<^esub>) b \<in> A"
up_to_iff_Ico: fixes J :: "nat set"   and n :: "nat" assumes "finite J" shows "(up_to J \<le> n) = (J \<subseteq> {0..<n})"
measurable_count_space_eq1: fixes f :: "'c \<Rightarrow> 'd"   and A :: "'c set"   and M :: "'d measure" shows "(f \<in> count_space A \<rightarrow>\<^sub>M M) = (f \<in> A \<rightarrow> space M)"
poly_altdef: fixes p :: "'b poly"   and x :: "'b" shows "poly p x = (\<Sum>i\<le>degree p. coeff p i * x ^ i)"
IN_h: fixes x :: "'v" shows "d_IN h x = (if reachable x then d_IN g x else 0)"
conjunctiveD: fixes x :: "'c \<Rightarrow> 'd"   and y :: "'c"   and z :: "'c" assumes "x \<in> Apply.conjunctive" shows "x (inf y z) = inf (x y) (x z)"
assms_1: shows "local_ring A addA multA zeroA oneA"
rep_eq: fixes x :: "int" shows "nat x = (case Rep_Integ x of (x, xa) \<Rightarrow> x - xa)"
singleton_union_fset_left: fixes a :: "'b"   and S :: "'b fset" shows "{|a|} |\<union>| S = insert_fset a S"
nomatch: fixes \<gamma> :: "('c \<Rightarrow> 'd \<Rightarrow> ternaryvalue) \<times> (action \<Rightarrow> 'd \<Rightarrow> bool)"   and m :: "'c match_expr"   and a :: "action"   and p :: "'d" assumes "\<not> Matching_Ternary.matches \<gamma> m a p" shows "\<gamma>,p\<turnstile> \<langle>[Rule m a], Undecided\<rangle> \<Rightarrow>\<^sub>\<alpha> Undecided"
leD: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d" assumes "class.order less_eq less"   and "less_eq y x" shows "\<not> less x y"
simps_1: fixes uu :: "'d list"   and u :: "'c \<Rightarrow> 'd" shows "set_clocks [] uu u = u"
f3: shows "r \<otimes> r' = r' \<otimes> r"
omega_fix: fixes x :: "'a" shows "x ^ \<omega> = x * x ^ \<omega> \<sqinter> (1::'a)"
LCons_Lazy_llist: fixes x :: "'a"   and xs :: "'a llist" shows "LCons x xs = Lazy_llist (\<lambda>_. Some (x, xs))"
prefixes_rev: fixes xs :: "'c list" shows "prefixes (rev xs) = map rev (suffixes xs)"
poly_exp_add: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and p :: "'b list"   and n :: "nat"   and d :: "nat"   and x :: "'b" assumes "class.comm_semiring_1 times one plus zero" shows "semiring_0.poly plus zero times (semiring_1.pexp one times plus zero p (n + d)) x = semiring_0.poly plus zero times (semiring_0.pmult plus zero times (semiring_1.pexp one times plus zero p n) (semiring_1.pexp one times plus zero p d)) x"
not_eq_order_implies_strict: fixes a :: "'b list"   and b :: "'b list" assumes "a \<noteq> b"   and "a \<le>s b" shows "a <s b"
insertion_trivial: fixes p :: "'a mpoly" shows "insertion (\<lambda>_. 0::'a) p = MPoly_Type.coeff p 0"
simps_6: fixes v :: "Ferrack.num" shows "not (Gt v) = NOT (Gt v)"
le_neq_implies_less: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n"   and "m \<noteq> n" shows "m < n"
cmp_map_def: fixes cmpk :: "'c \<Rightarrow> 'c \<Rightarrow> comp_res"   and cmpv :: "'d \<Rightarrow> 'd \<Rightarrow> comp_res" shows "cmp_map cmpk cmpv \<equiv> cmp_extend (Collect (finite \<circ> dom)) (cmp_img (linorder.sorted_list_of_map (comp2le cmpk)) (cmp_lex (cmp_prod cmpk cmpv)))"
G_def: shows "G \<equiv> \<lparr>g_V = UNIV, g_E = step, g_V0 = Collect init\<rparr>"
res_lim_Zp_cauchy_0_2: fixes p :: "int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and f :: "int"   and k :: "nat" assumes "padic_integers p"   and "padic_integers.is_Zp_cauchy p s"   and "f = res_lim s k" shows "f \<in> carrier (residue_ring (p ^ k))"
max_bot: fixes x :: "'b" shows "max (0::'b) x = x"
p_def: shows "p = (\<lambda>s\<in>S. P_sup s (\<lambda>\<omega>. (HLD S1 suntil HLD S2) (s ## \<omega>)))"
wf_def: fixes r :: "('d \<times> 'd) set" shows "wf r = (\<forall>P. (\<forall>x. (\<forall>y. (y, x) \<in> r \<longrightarrow> P y) \<longrightarrow> P x) \<longrightarrow> (\<forall>x. P x))"
root_def: fixes n :: "nat"   and x :: "real" shows "root n x = (if n = 0 then 0 else the_inv (\<lambda>y. sgn y * \<bar>y\<bar> ^ n) x)"
gallery_preimage: fixes Cs :: "'a set list" assumes "order.greater_eq (folding.\<C> - f \<turnstile> folding.\<C>) (set Cs)"   and "folding.gallery (f \<Turnstile> Cs)" shows "folding.gallery Cs"
simps_1: fixes uu :: "'a extN" shows "(extN.Bot \<le> uu) = True"
errMOD_igFreshIGOpSTR: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) model" assumes "igOpIPresIGWls MOD"   and "igFreshIGOp MOD" shows "igFreshIGOpSTR (errMOD MOD)"
qAFreshAll_imp_qFreshAll: fixes xs :: "'a"   and x :: "'b"   and X :: "('c, 'd, 'a, 'b, 'e) qTerm"   and A :: "('c, 'd, 'a, 'b, 'e) qAbs" shows "(qAFresh xs x X \<longrightarrow> qFresh xs x X) \<and> (qAFreshAbs xs x A \<longrightarrow> qFreshAbs xs x A)"
prime_nat_main: fixes ni :: "nat \<times> nat \<times> nat list"   and n :: "nat"   and i :: "nat"   and "is" :: "nat list"   and jj :: "nat"   and res :: "bool" assumes "ni = (n, i, is)"   and "2 \<le> i"   and "2 \<le> n"   and "\<And>j. \<lbrakk>2 \<le> j; j < i\<rbrakk> \<Longrightarrow> \<not> j dvd n"   and "\<And>j. \<lbrakk>i \<le> j; j < jj; prime j\<rbrakk> \<Longrightarrow> j \<in> set is"   and "i \<le> jj"   and "sorted is"   and "distinct is"   and "candidate_invariant jj"   and "set is \<subseteq> {i..<jj}"   and "res = prime_nat_main n jj is" shows "res = prime n"
gi: fixes i :: "nat" shows "g i \<subseteq> g (Suc i)"
integral_measure_pmf_real: fixes A :: "'e set"   and M :: "'e pmf"   and f :: "'e \<Rightarrow> real" assumes "finite A"   and "\<And>a. \<lbrakk>a \<in> set_pmf M; f a \<noteq> 0\<rbrakk> \<Longrightarrow> a \<in> A" shows "measure_pmf.expectation M f = (\<Sum>a\<in>A. f a * pmf M a)"
fls_right_inverse_delta: fixes b :: "'a"   and a :: "int"   and x :: "'a" assumes "b \<noteq> (0::'a)" shows "fls_right_inverse (Abs_fls (\<lambda>n. if n = a then b else (0::'a))) x = Abs_fls (\<lambda>n. if n = - a then x else (0::'a))"
iMin_atLeastLessThan: fixes m :: "'b"   and n :: "'b" assumes "m < n" shows "iMin {m..<n} = m"
order_iff: fixes a :: "'a"   and b :: "'a" shows "(a \<preceq> b) = (a = a \<sqinter> b)"
sopen_sclose_eq_ssubst: fixes sa :: "char list"   and pa :: "char list"   and p :: "sterm"   and t :: "sterm"   and n :: "nat"   and s :: "sterm" assumes "sa \<noteq> pa"   and "sa \<notin> FV p"   and "lc t" shows "{n \<rightarrow> [s,p]} {n \<leftarrow> [sa,pa]} t = [sa \<rightarrow> s] [pa \<rightarrow> p] t"
arcs_dv: shows "arcs (del_vert u) = arcs G"
vfsequence_vcard_in_omega: shows "vcard [a, a]\<^sub>\<circ> \<in>\<^sub>\<circ> \<omega>"
tail_alt: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" shows "punit.tail p = except p {punit.lt p}"
eq_inf_range: fixes w\<^sub>1 :: "'action stream"   and w\<^sub>2 :: "'action stream" assumes "w\<^sub>1 =\<^sub>I w\<^sub>2" shows "sset w\<^sub>1 = sset w\<^sub>2"
tranclp_on_imp_P: fixes U :: "'c \<Rightarrow> bool"   and r :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and x :: "'c"   and y :: "'c" assumes "on U r\<^sup>+\<^sup>+ x y" shows "U x \<and> U y"
Loeb_Form_axioms: shows "Loeb_Form var trm fmla Var num FvarsT substT Fvars subst eql cnj imp all exi prv bprv enc S P"
index_of_c_fst_main: shows "nat_to_pr index_of_c_fst = c_fst"
simplex_ns_unsat: fixes cs :: "('i \<times> 'a ns_constraint) list"   and I :: "'i list" assumes "solve_exec_ns cs = Inl I" shows "minimal_unsat_core_ns (set I) (set cs)"
sim_first2: fixes s :: "nat \<Rightarrow> 'a"   and t :: "nat \<Rightarrow> 'a" assumes "s \<approx> t" shows "s 0 = t 0"
of_nat_take_bit: fixes plus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and minus :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and zero :: "'k"   and times :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and one :: "'k"   and divide :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and modulo :: "'k \<Rightarrow> 'k \<Rightarrow> 'k"   and bit :: "'k \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and drop_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and take_bit :: "nat \<Rightarrow> 'k \<Rightarrow> 'k"   and n :: "nat"   and m :: "nat" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "semiring_1.of_nat one plus zero (take_bit n m) = take_bit n (semiring_1.of_nat one plus zero m)"
1: fixes u :: "'t"   and v :: "'t" assumes "map_component ((+) i) u = map_component ((+) i) v" shows "u = v"
prv_exi_imp: fixes x :: "'var"   and \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "x \<in> var"   and "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "x \<notin> Fvars \<chi>"   and "prv (all x (imp \<phi> \<chi>))" shows "prv (imp (exi x \<phi>) \<chi>)"