hcju/isabelleps huggingface dataset:
simps_1: fixes P :: "jvm_prog"   and h :: "heap"   and stk :: "val list"   and loc :: "val list"   and C :: "char list"   and M :: "char list"   and pc :: "nat"   and C' :: "char list"   and Cs :: "char list list"   and frs :: "frame list"   and sh :: "sheap" shows "exec_step P h stk loc C M pc (Calling C' Cs) frs sh = exec_Calling C' Cs P h stk loc C M pc frs sh"
eq: shows "f \<equiv> C (heap.fixp_fun (\<lambda>f. U (F (C f))))"
SETcod: fixes A :: "ZF"   and B :: "ZF"   and f :: "ZF \<Rightarrow> ZF" assumes "isZFfun (ZFfun A B f)" shows "cod\<^bsub>SET\<^esub> ZFfun A B f = B"
lm06: fixes f :: "nat \<Rightarrow> 'c"   and P :: "nat \<Rightarrow> bool"   and m :: "nat" shows "map f (concat (map (\<lambda>n. if P n then [n] else []) [0..<m])) = concat (map (\<lambda>n. if P n then [f n] else []) [0..<m])"
\<tau>exec_meth_def: fixes ci :: "('addr, 'heap) check_instr"   and P :: "(nat \<times> nat \<times> 'addr instr list \<times> (nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list) prog"   and ins :: "'addr instr list"   and xt :: "(nat \<times> nat \<times> String.literal option \<times> nat \<times> nat) list"   and t :: "'thread_id"   and h :: "'heap"   and s :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option"   and s' :: "'addr val list \<times> 'addr val list \<times> nat \<times> 'addr option" shows "\<tau>exec_meth ci P ins xt t h s s' = (exec_meth ci P ins xt t h s \<lbrace>\<rbrace> h s' \<and> (snd (snd (snd s)) = None \<longrightarrow> \<tau>instr P h (fst s) (ins ! fst (snd (snd s)))))"
triv_pref: fixes r :: "'b list"   and s :: "'b list" shows "r \<le>p r \<cdot> s"
bet__ts: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and Y :: "'a"   and B :: "'a"   and X :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "A \<noteq> Y"   and "\<not> Tarski_neutral_dimensionless.Col Bet A B X"   and "Bet X A Y" shows "Tarski_neutral_dimensionless.TS Bet A B X Y"
AssertBoundNoLhs_axioms: shows "AssertBoundNoLhs assert_bound"
assms_3: shows "val_Zp_dist a b = \<infinity>"
cindex_pathE_reversepath': fixes g :: "real \<Rightarrow> complex"   and z :: "complex" shows "cindex_pathE g z = - cindex_pathE (reversepath g) z"
functionCompletion: fixes x :: "'c"   and f :: "('c \<times> 'd) set" assumes "x \<in> Domain f" shows "toFunction f x = (f Elsee (0::'d)) x"
subst_typ_fv: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and B :: "Core.term" shows "apsnd (subst_typ insts) ` fv B = fv (subst_typ' insts B)"
val_eq_n_val_eq: fixes v :: "'a \<Rightarrow> ant"   and x :: "'a"   and y :: "'a" assumes "valuation K v"   and "x \<in> carrier K"   and "y \<in> carrier K" shows "(v x = v y) = (n_val K v x = n_val K v y)"
real_alg_of_real_def: fixes x :: "real" shows "real_alg_of_real x = (if \<exists>y. x = real_of y then THE y. x = real_of y else 0)"
elims_3: fixes x :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" assumes "\<not> valid_chain_list x" obtains oneC :: "int \<times> (real \<Rightarrow> real \<times> real)" and v :: "int \<times> (real \<Rightarrow> real \<times> real)" and va :: "(int \<times> (real \<Rightarrow> real \<times> real)) list" where "x = oneC # v # va" and "\<not> (pathfinish (coeff_cube_to_path oneC) = pathstart (rec_join (v # va)) \<and> valid_chain_list (v # va))"
"1_3": shows "(ptr, child) \<in> (parent_child_rel h)\<^sup>*"
cut_less_Int: fixes A :: "'b set"   and B :: "'b set"   and t :: "'b" shows "(A \<inter> B) \<down>< t = A \<down>< t \<inter> B \<down>< t"
distinct_55: fixes x31 :: "char list"   and x32 :: "expr"   and x4 :: "val" shows "\<lparr>x31\<rparr>x32 \<noteq> Val x4"
interchange_columns_code_nth: fixes A :: "(('c, 'd) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and a :: "'d"   and b :: "'d" shows "($h) (interchange_columns A a b) = interchange_columns_row A a b"
id_rad_ideal: fixes I :: "'a set" assumes "ideal R I" shows "ideal R (rad_ideal R I)"
lincomb_union2: fixes A :: "'c set"   and B :: "'c set"   and a :: "'c \<Rightarrow> 'a"   and b :: "'c \<Rightarrow> 'a" assumes "finite (A \<union> B)"   and "A \<union> B \<subseteq> carrier M"   and "A \<inter> B = {}"   and "a \<in> A \<rightarrow> carrier R"   and "b \<in> B \<rightarrow> carrier R" shows "lincomb a A \<oplus>\<^bsub>M\<^esub> lincomb b B = lincomb (\<lambda>v. if v \<in> A then a v else b v) (A \<union> B)"
subcring_def: fixes H :: "'c set"   and R :: "('c, 'd) ring_scheme" shows "subcring H R \<equiv> subring H R \<and> subcring_axioms H R"
Arbitrary_Except2D: fixes v :: "('a \<times> bool list) list"   and n :: "nat"   and r :: "nat"   and s :: "nat" assumes "v \<in> lang n (Atom (Arbitrary_Except2 r s))"   and "r < n"   and "s < n" shows "\<exists>x. v = [x] \<and> snd x ! r \<and> snd x ! s"
adds_antisym: fixes s :: "'b"   and t :: "'b" assumes "s adds t"   and "t adds s" shows "s = t"
OclAnd_assoc: fixes X :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Y :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and Z :: "'a state \<times> 'a state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>" shows "(X and (Y and Z)) = (X and Y and Z)"
lfp_lowerbound: fixes a :: "'a"   and f :: "'a \<Rightarrow> 'a" assumes "a \<in> L"   and "f a \<le> a" shows "local.lfp f \<le> a"
uniq: fixes x :: "'a"   and y :: "real"   and z :: "real" assumes "(x, y) \<in> g"   and "(x, z) \<in> g" shows "z = y"
GreekMortal: fixes Mortal :: "'a set"   and Human :: "'a set"   and Greek :: "'a set" assumes "Mortal Q Human"   and "Human Q Greek" shows "Mortal Q Greek"
intros: fixes a :: "'g"   and b :: "'h"   and r :: "('g \<times> 'h) set" assumes "(a, b) \<in> r" shows "(b, a) \<in> r\<inverse>"
simps_2: fixes rel :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and v :: "'d"   and va :: "'d list" shows "merge_wrt rel [] (v # va) = v # va"
nths_reencode_eq: fixes B :: "nat set"   and A :: "nat set"   and xs :: "'a list" assumes "B \<subseteq> A" shows "nths (nths xs A) (ind_in_set A ` B) = nths xs B"
decidable_complement: fixes X :: "nat set" assumes "decidable X" shows "decidable (- X)"
extendf_deriv: fixes f' :: "('sf + 'sz, 'sz) trm"   and I :: "('sf, 'sc, 'sz) interp"   and \<nu> :: "(real, 'sz) vec \<times> (real, 'sz) vec"   and i_f :: "'sf" assumes "dfree f'"   and "is_interp I" shows "\<exists>f''. \<forall>x. ((\<lambda>R. dterm_sem (extendf I R) f' \<nu>) has_derivative extendf_deriv I i_f f' \<nu> x) (at x)"
simps_5: fixes r :: "mregex"   and s :: "mregex" shows "LPD (MTimes r s) = MTimesR (LPD r) s \<union> LPD s"
siso_transC: fixes c :: "('test, 'atom) com"   and s :: "'state"   and c' :: "('test, 'atom) com"   and s' :: "'state" assumes "siso c"   and "(c, s) \<rightarrow>c (c', s')" shows "siso c'"
BR: fixes a :: "subject"   and y :: "name"   and Q' :: "pi" assumes "Q \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> Q'"   and "x \<sharp> a"   and "x \<noteq> y"   and "y \<sharp> C" shows "\<exists>P'. P \<longmapsto> a\<guillemotleft>y\<guillemotright> \<prec> P' \<and> derivative P' (<\<nu>x>Q') a y Rel"
CM3_k_parts_knows_Spy: fixes evs :: "event list"   and A :: "agent"   and S :: "agent"   and AO :: "nat"   and K :: "nat"   and R :: "agent"   and hs :: "msg" assumes "evs \<in> certified_mail"   and "Notes TTP \<lbrace>Agent A, Agent TTP, Crypt (pubK TTP) \<lbrace>Agent S, Number AO, Key K, Agent R, hs\<rbrace>, Key (RPwd R), hs\<rbrace> \<in> set evs" shows "Key K \<in> parts (knows Spy evs)"
le_Javatype_def: fixes A :: "Javatype"   and B :: "Javatype" shows "A \<le> B \<equiv> A \<preceq> B"
post_fst_k_def: fixes k :: "nat" shows "post_fst_k k = mat N N (\<lambda>(i, j). if i = j \<and> f i \<and> i < k then 1 else 0)"
vpath_def: fixes p :: "'a list"   and G :: "('a, 'b) pre_digraph" shows "vpath p G \<equiv> vwalk p G \<and> distinct p"
\<gamma>_1: fixes m :: "nat" shows "gauge (\<gamma> m)"
eq_vector_fraction_iff: fixes x :: "'c"   and u :: "real"   and v :: "real"   and a :: "'c" shows "(x = (u / v) *\<^sub>R a) = (if v = 0 then x = (0::'c) else v *\<^sub>R x = u *\<^sub>R a)"
vars_of_replacement: fixes x :: "'b"   and s :: "'b trm"   and t :: "'b trm"   and p :: "indices list"   and v :: "'b trm" assumes "x \<in> vars_of s"   and "replace_subterm t p v s" shows "x \<in> vars_of t \<union> vars_of v"
marked_imp_black_or_grey_2: fixes r :: "'e"   and s :: "'f process_name \<Rightarrow> ('g, 'f, 'h, 'e) Model.local_state" assumes "\<not> white r s" shows "\<not> obj_at (\<lambda>s. True) r s \<or> black r s \<or> grey r s"
zero_less_ceiling: fixes x :: "'a" shows "(0 < \<lceil>x\<rceil>) = ((0::'a) < x)"
ids_def: fixes t :: "'a" shows "ids t = frees t |\<union>| consts t"
at_most_one_mset_mset_diff: fixes a :: "'c"   and M :: "'c multiset" assumes "a \<notin># remove1_mset a M" shows "set_mset (remove1_mset a M) = set_mset M - {a}"
simps_6: fixes \<mu> :: "'a ltln"   and \<mu>' :: "'a ltln" shows "pure_eventual (\<mu> R\<^sub>n \<mu>') = (pure_eventual \<mu> \<and> pure_eventual \<mu>')"
simps_3: fixes f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> bool"   and kx :: "'a"   and vx :: "'d"   and xs :: "('a \<times> 'd) list" shows "prod_ord_pair f ((kx, vx) # xs) [] = (f kx vx (0::'e) \<and> prod_ord_pair f xs [])"
intros_28: fixes vs :: "e list"   and n :: "nat"   and t1s :: "t list"   and t2s :: "t list"   and m :: "nat"   and es :: "b_e list" assumes "const_list vs"   and "length vs = n"   and "length t1s = n"   and "length t2s = m" shows "\<lparr>vs @ [$Loop (t1s _> t2s) es]\<rparr> \<leadsto> \<lparr>[Label n [$Loop (t1s _> t2s) es] (vs @ ($* es))]\<rparr>"
range_composition: fixes f :: "'c \<Rightarrow> 'a"   and g :: "'b \<Rightarrow> 'c" shows "range (\<lambda>x. f (g x)) = f ` range g"
f_Exec_Stream_Acc_LocalState_Cons: fixes k :: "nat"   and localState :: "'e \<Rightarrow> 'd"   and trans_fun :: "'f message_af \<Rightarrow> 'e \<Rightarrow> 'e"   and x :: "'f message_af"   and xs :: "'f message_af list"   and c :: "'e" assumes "0 < k" shows "f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun (x # xs) c = localState (f_Exec_Comp trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) c) # f_Exec_Comp_Stream_Acc_LocalState k localState trans_fun xs (f_Exec_Comp trans_fun (x # NoMsg\<^bsup>k - Suc 0\<^esup>) c)"
F1set2_JF1set_JF1set: fixes a' :: "'d JF1"   and a :: "'d JF1"   and c :: "'d" assumes "a' \<in> F1set2 (dtor1 a)"   and "c \<in> JF1set a'" shows "c \<in> JF1set a"
HPhase1or2ReadThen_HInv4b_q: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and r :: "Proc"   and q :: "Proc" assumes "HPhase1or2ReadThen s s' p d r"   and "HInv4b s q"   and "p \<noteq> q" shows "HInv4b s' q"
simps_3: fixes v :: "'b"   and va :: "('a, 'b) Digit"   and vb :: "('a, 'b) FingerTreeStruc"   and vc :: "('a, 'b) Digit"   and xs :: "('a, 'b) Node list" shows "FingerTreeStruc.app3 (Deep v va vb vc) xs Empty = FingerTreeStruc.rconsNlist (Deep v va vb vc) xs"
block_complement_subset_points: fixes ps :: "'a set"   and bl :: "'a set" assumes "ps \<subseteq> source.block_complement bl" shows "ps \<subseteq> \<V>"
distinct_5: fixes x41 :: "expr"   and x42 :: "com"   and x43 :: "com" shows "Skip \<noteq> if (x41) x42 else x43"
below_set_def: shows "(\<sqsubseteq>) = (\<subseteq>)"
lossless_protocol_14_OT: fixes S1_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> unit \<Rightarrow> 'a spmf"   and R1_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> 'a spmf"   and S2_14_OT :: "bool \<times> bool \<Rightarrow> bool \<Rightarrow> 'b spmf"   and R2_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> 'b spmf"   and protocol_14_OT :: "bool \<times> bool \<times> bool \<times> bool \<Rightarrow> bool \<times> bool \<Rightarrow> (unit \<times> bool) spmf"   and adv_14_OT :: "real"   and S :: "bool \<times> bool \<times> bool \<times> bool"   and C :: "bool \<times> bool" assumes "gmw_base S1_14_OT R1_14_OT S2_14_OT R2_14_OT protocol_14_OT adv_14_OT" shows "lossless_spmf (protocol_14_OT S C)"
proper_it_def: fixes it :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'a \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'a"   and it' :: "('c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c" shows "proper_it it it' \<equiv> \<exists>l. it = foldli l \<and> it' = foldli l"
holomorphic_rGamma: fixes A :: "complex set" shows "rGamma holomorphic_on A"
convergent_prod_Suc_iff: fixes f :: "nat \<Rightarrow> 'a" shows "convergent_prod (\<lambda>n. f (Suc n)) = convergent_prod f"
not_cylce_free_dest: fixes m :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and n :: "nat" assumes "\<not> cycle_free m n" shows "\<exists>k\<le>n. \<not> cycle_free_up_to m k n"
can_extend_acyclic_order_to_strict_linear: fixes base_r :: "('b \<times> 'b) set" assumes "acyclic base_r" shows "\<exists>r. strict_linear_order r \<and> base_r \<subseteq> r"
valid_prefix_def: fixes pf :: "'a prefix_match" shows "valid_prefix pf = (pfxm_mask pf && pfxm_prefix pf = 0)"
rel_inject_1: fixes R1 :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and R2 :: "'c \<Rightarrow> 'd \<Rightarrow> bool"   and x1 :: "'e \<Rightarrow> ('e, 'a, 'c) sp\<^sub>\<mu>"   and y1 :: "'e \<Rightarrow> ('e, 'b, 'd) sp\<^sub>\<mu>" shows "rel_sp\<^sub>\<mu> R1 R2 (Get x1) (Get y1) = rel_fun (=) (rel_sp\<^sub>\<mu> R1 R2) x1 y1"
sim_def: fixes M :: "('b, 'c, 'd) KripkeStructure"   and M' :: "('b, 'c, 'e) KripkeStructure"   and f :: "'d \<Rightarrow> 'e" shows "sim M M' f \<equiv> sim_range M M' f \<and> sim_val M M' f \<and> sim_f M M' f \<and> sim_r M M' f"
generate_is_subgroup: fixes H :: "'a set" assumes "H \<subseteq> carrier (G\<lparr>carrier := generate G {a}\<rparr>)" shows "subgroup (generate (G\<lparr>carrier := generate G {a}\<rparr>) H) (G\<lparr>carrier := generate G {a}\<rparr>)"
dvd_add: fixes times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.comm_semiring_1 times one plus zero"   and "dvd.dvd times a b"   and "dvd.dvd times a c" shows "dvd.dvd times a (plus b c)"
Nil: fixes M\<^sub>0 :: "('fun, 'var) Term.term set"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term" shows "sem\<^sub>e\<^sub>s\<^sub>t_d M\<^sub>0 \<I> []"
emptyLeftNonSync: fixes A :: "'a event set"   and t :: "'a event list" shows "\<forall>s. s setinterleaves (([], t), A) \<longrightarrow> (\<forall>t1. t1 \<in> set t \<longrightarrow> t1 \<notin> A)"
Le_def: shows "Le \<equiv> {(x, y). x \<le> y}"
A: shows "(\<exists>s a b d. f = d * (s + a * b) \<and> NonceS (S (Card n, n, run)) = Some s \<and> IntMapK (S (Card n, n, run)) = Some b \<and> ExtMapK (S (Card n, n, run)) = Some a \<and> IntAgrK (S (Card n, n, run)) = Some d \<and> d \<noteq> 0 \<and> s + a * b \<noteq> 0) \<or> (\<exists>b. Pri_AgrK b \<in> A \<and> ExtMapK (S (User m, n, run)) = Some b)"
atMost_borel: fixes a :: "'a" shows "{..a} \<in> sets borel"
neg_conj_cancel1: fixes P :: "'a upred"   and Q :: "'a upred" shows "(\<not> P \<and> (P \<or> Q)) = (\<not> P \<and> Q)"
subset_iff: fixes A :: "'c set"   and B :: "'c set" shows "(A \<subseteq> B) = (\<forall>t. t \<in> A \<longrightarrow> t \<in> B)"
reducible_nonredundant_imp_minimal: fixes g :: "'g" assumes "old.reducible g"   and "\<not> notriv.redundant g" shows "notriv.cytronMinimal g"
convex_same_rel_interior_closure_straddle: fixes S :: "'c set"   and T :: "'c set" assumes "convex S"   and "convex T" shows "(rel_interior S = rel_interior T) = (rel_interior S \<subseteq> T \<and> T \<subseteq> closure S)"
cqt_further_11: fixes v :: "i"   and \<phi> :: "'a \<Rightarrow> \<o>"   and \<psi> :: "'a \<Rightarrow> \<o>" shows "[(\<^bold>\<forall>\<alpha>. \<phi> \<alpha>) \<^bold>& (\<^bold>\<forall>\<alpha>. \<psi> \<alpha>) \<^bold>\<rightarrow> (\<^bold>\<forall>\<alpha>. \<phi> \<alpha> \<^bold>\<equiv> \<psi> \<alpha>) in v]"
better_Cut_substc2: fixes x :: "name"   and y :: "name"   and P :: "trm"   and b :: "coname"   and a :: "coname"   and M :: "trm"   and N :: "trm" assumes "x \<sharp> (y, P)"   and "b \<sharp> (a, M)"   and "N \<noteq> Ax x b" shows "Cut <a>.M x.N{b:=y.P} = Cut <a>.M x.N{b:=y.P}"
prts: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and a :: "'b" assumes "class.lattice_ab_group_add plus zero minus uminus less_eq less inf sup" shows "a = plus (lattice_ab_group_add.pprt zero sup a) (lattice_ab_group_add.nprt zero inf a)"
char_poly_hom: fixes A :: "real mat"   and n :: "nat" assumes "A \<in> carrier_mat n n" shows "char_poly (of_real_hom.mat_hom A) = map_poly of_real (char_poly A)"
insert_is_Un: fixes a :: "'b"   and A :: "'b set" shows "insert a A = {a} \<union> A"
f0: fixes F :: "nat \<Rightarrow> 'b poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'b"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'b"   and i :: "nat" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>"   and "i \<noteq> 0" shows "(f i = (0::'b)) = (k < i)"
deg_shifts_def: fixes X :: "'x set"   and d :: "nat"   and fs :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) list" assumes "finite X" shows "pp_pm.deg_shifts X d fs = concat (map (\<lambda>f. map (\<lambda>t. MPoly_Type_Class.punit.monom_mult (1::'b) t f) (pp_pm.punit.pps_to_list (deg_le_sect X (d - poly_deg f)))) fs)"
extended_Ax_prems_empty: fixes r :: "'b sequent list \<times> 'b sequent"   and S :: "'b sequent" assumes "r \<in> Ax" shows "fst (extendRule S r) = []"
gbinomial_int_mult_fact': fixes a :: "int"   and k :: "nat" shows "(a gchoose k) * fact k = (\<Prod>i = 0..<k. a - int i)"
extend_f_def: fixes f :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool"   and g :: "partition_id_t \<Rightarrow> partition_id_t \<Rightarrow> bool" shows "extend_f f g \<equiv> \<lambda>p1 p2. f p1 p2 \<or> g p1 p2"
nex: shows "\<not> (\<exists>k>x. k < y \<and> poly [:c, b, a:] k = 0)"
has_field_derivative_stirling_sum'_complex: fixes j :: "nat"   and x :: "complex"   and m :: "nat" assumes "0 < j"   and "0 < Re x" shows "(stirling_sum' j m has_field_derivative stirling_sum' (Suc j) m x) (at x)"
Urysohn_local: fixes U :: "'c set"   and S :: "'c set"   and T :: "'c set"   and a :: "'d"   and b :: "'d" assumes "closedin (top_of_set U) S"   and "closedin (top_of_set U) T"   and "S \<inter> T = {}" obtains f :: "'c \<Rightarrow> 'd" where "continuous_on U f" and "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a b" and "\<And>x. x \<in> S \<Longrightarrow> f x = a" and "\<And>x. x \<in> T \<Longrightarrow> f x = b"
isolated_singularity_at_id: fixes z :: "complex" shows "isolated_singularity_at (\<lambda>w. w) z"
base_2: fixes n :: "nat" shows "x n \<in> sfis (u n) M"
wf_implies_termi_reach: fixes \<Gamma> :: "'b \<Rightarrow> ('a, 'b, 'c) com option"   and c :: "('a, 'b, 'c) com"   and s :: "('a, 'c) xstate"   and cfg1 :: "('a, 'b, 'c) com \<times> ('a, 'c) xstate"   and c1 :: "('a, 'b, 'c) com"   and s1 :: "('a, 'c) xstate" assumes "wf {(cfg2, cfg1). \<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* cfg1 \<and> \<Gamma>\<turnstile> cfg1 \<rightarrow> cfg2}"   and "\<Gamma>\<turnstile> (c, s) \<rightarrow>\<^sup>* cfg1"   and "cfg1 = (c1, s1)" shows "\<Gamma>\<turnstile>c1 \<down> s1"
state_cont: fixes cfg :: "'a cfg"   and s :: "'a" shows "state (cont cfg s) = s"
reduce_less_plus_consts': fixes n2 :: "nat"   and n1 :: "nat"   and x :: "nat"   and y :: "nat" assumes "n2 \<le> n1"   and "x + n1 < y + n2" shows "x + (n1 - n2) < y"
mtf_MTF: fixes n :: "nat"   and s :: "'a list" assumes "n < length rs"   and "length s = length s0" shows "mtf (rs ! n) s = step s (rs ! n) (off MTF s0 rs ! n)"
bij_simp: fixes x :: "'a"   and y :: "'a" shows "(f x = f y) = (x = y)"
assms_2: shows "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"
elims: fixes x :: "'heap"   and xa :: "'addr val"   and y :: "ty option" assumes "typeof\<^bsub>x\<^esub> xa = y" obtains h :: "'heap" and a :: "'addr" where "x = ha__" and "xa = Addr a" and "y = map_option ty_of_htype (typeof_addr ha__ a)"   | h :: "'heap" where "x = hb__" and "xa = Unit" and "y = typeof Unit"   | h :: "'heap" where "x = hc__" and "xa = Null" and "y = typeof Null"   | h :: "'heap" and va :: "bool" where "x = hd__" and "xa = Bool va__" and "y = typeof (Bool va__)"   | h :: "'heap" and va :: "32 word" where "x = h" and "xa = Intg va" and "y = typeof (Intg va)"
mk_lhsv_thm_1: fixes c :: "com"   and cmd :: "com"   and \<pi> :: "char list \<Rightarrow> com option" assumes "c \<equiv> cmd" shows "lhsv \<pi> c = ANALYZE (lhsv \<pi> cmd)"
intro: fixes L :: "('c, 'd) gorder_scheme" assumes "weak_upper_semilattice L"   and "weak_lower_semilattice L" shows "weak_lattice L"
encode_interfering_operator_exclusion_def: fixes \<Pi> :: "'a strips_problem"   and t :: "nat" shows "encode_interfering_operator_exclusion \<Pi> t \<equiv> let ops = \<Pi>\<^sub>\<O>; interfering = filter (\<lambda>(op\<^sub>1, op\<^sub>2). index ops op\<^sub>1 \<noteq> index ops op\<^sub>2 \<and> are_operators_interfering op\<^sub>1 op\<^sub>2) (List.product ops ops) in foldr (\<^bold>\<and>) (concat (map (\<lambda>(op\<^sub>1, op\<^sub>2). map (\<lambda>k. encode_interfering_operator_pair_exclusion \<Pi> k op\<^sub>1 op\<^sub>2) [0..<t]) interfering)) (\<^bold>\<not> \<bottom>)"
map_of_inject_set: fixes xs :: "('a \<times> 'b) list"   and ys :: "('a \<times> 'b) list" assumes "distinct (map fst xs)"   and "distinct (map fst ys)" shows "(map_of xs = map_of ys) = (set xs = set ys)"
intro: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" assumes "comp_fun_commute f"   and "comp_fun_idem_axioms f" shows "comp_fun_idem f"
simps_2: fixes y :: "'a" shows "flat_1 (allow (deny y)) = allow y"
assms: shows "set TI' = {(a, b). (a, b) \<in> (set TI)\<^sup>+ \<and> a \<noteq> b}"
stable_on_blocking_onD: fixes ds :: "'b set"   and X :: "'a set"   and X'' :: "'a set"   and h :: "'c" assumes "Bossy.stable_on ds X"   and "X'' = BCh h (X \<union> X'')"   and "X'' \<subseteq> Bossy.CD_on ds (X \<union> X'')" shows "X'' = BCh h X"
simps_2: fixes xs :: "'c list"   and opt :: "('a, 'b, 'c) env option"   and b :: "'b"   and es :: "'c \<Rightarrow> ('a, 'b, 'c) env option" shows "update xs opt (Env b es) = (case xs of [] \<Rightarrow> case opt of None \<Rightarrow> Env b es | Some e \<Rightarrow> e | y # ys \<Rightarrow> Env b (es(y := ??.Nested_Environment.update_option ys opt (es y))))"
fmlookup_fmrestrict_set_dom: fixes vs :: "'d set"   and s :: "('d, 'e) fmap" shows "dom (fmlookup (fmrestrict_set vs s)) = vs \<inter> fmdom' s"
run_sim2: fixes G :: "('a, 'b) graph_rec_scheme"   and f :: "'a \<Rightarrow> 'c"   and ecnv :: "('a, 'b) graph_rec_scheme \<Rightarrow> 'd"   and r :: "nat \<Rightarrow> 'c" assumes "g_rename_precond G f"   and "graph_defs.is_run (fr_rename_ext ecnv f G) r" shows "graph_defs.is_run G (g_rename_precond.fi G f \<circ> r)"
fold_mset_add_mset: fixes f :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s :: "'e"   and x :: "'d"   and M :: "'d multiset" assumes "comp_fun_commute f" shows "fold_mset f s (add_mset x M) = f x (fold_mset f s M)"
simps: fixes var :: "nat"   and L :: "atom list"   and F :: "atom fm list" shows "qe_eq_repeat var L F = (case luckyFind var L F of None \<Rightarrow> let (A, L') = find_eq var L in qe_eq_repeat_helper var A L' F | Some F \<Rightarrow> F)"
trans_Sup_of_chain: fixes C :: "('c \<times> 'c) set set" assumes "set_chain C"   and "\<And>R. R \<in> C \<Longrightarrow> trans R" shows "trans (\<Union> C)"