hcju/isabelleps huggingface dataset:
nucleus_alt_def1: fixes f :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b" assumes "nucleus f" shows "f x \<rightarrow> f y = x \<rightarrow> f y"
assms_2: fixes i :: "'x"   and j :: "'x" assumes "R i j" shows "i \<le> j"
AbstTerm_def: fixes v :: "hf"   and i :: "hf"   and x :: "hf"   and x' :: "hf" shows "AbstTerm v i x x' \<equiv> Ord i \<and> (\<exists>s k. SeqStTerm v (q_Ind i) x x' s k)"
le_funD: fixes f :: "'k \<Rightarrow> 'l"   and g :: "'k \<Rightarrow> 'l"   and x :: "'k" assumes "f \<le> g" shows "f x \<le> g x"
HPhase1or2Write_HInv4c: fixes s :: "state"   and s' :: "state"   and p :: "Proc"   and d :: "Disk"   and q :: "Proc" assumes "HPhase1or2Write s s' p d"   and "HInv4c s q"   and "Inv2c s" shows "HInv4c s' q"
bin_bl_bin: fixes n :: "nat"   and w :: "int" shows "bl_to_bin (bin_to_bl n w) = take_bit n w"
pdevs_val_filter_pdevs: fixes e :: "nat \<Rightarrow> real"   and I :: "nat \<Rightarrow> 'a \<Rightarrow> bool"   and x :: "'a pdevs" shows "pdevs_val e (filter_pdevs I x) = (\<Sum>i\<in>{..<degree x} \<inter> {i. I i (pdevs_apply x i)}. e i *\<^sub>R pdevs_apply x i)"
merge_F_cong: fixes x :: "('e, 'f, 'g, 'h) F\<^sub>m"   and ma :: "'e \<Rightarrow> 'e \<Rightarrow> 'e option"   and ma' :: "'e \<Rightarrow> 'e \<Rightarrow> 'e option"   and mb :: "'g \<Rightarrow> 'g \<Rightarrow> 'g option"   and mb' :: "'g \<Rightarrow> 'g \<Rightarrow> 'g option"   and rha :: "'e \<Rightarrow> 'f"   and rhb :: "'g \<Rightarrow> 'h"   and y :: "('e, 'f, 'g, 'h) F\<^sub>m" assumes "\<And>a b. a \<in> set1_F\<^sub>m x \<Longrightarrow> ma a b = ma' a b"   and "\<And>a b. a \<in> set3_F\<^sub>m x \<Longrightarrow> mb a b = mb' a b" shows "merge_F rha ma rhb mb x y = merge_F rha ma' rhb mb' x y"
fun_type_id_eq: fixes arity :: "'a \<Rightarrow> nat"   and public :: "'a \<Rightarrow> bool"   and Ana :: "('a, 'b) Term.term \<Rightarrow> ('a, 'b) Term.term list \<times> ('a, 'b) Term.term list"   and \<Gamma> :: "('a, 'b) Term.term \<Rightarrow> ('a, 'c) Term.term"   and f :: "'a"   and X :: "('a, 'b) Term.term list"   and g :: "'a"   and Y :: "('a, 'c) Term.term list" assumes "typed_model arity public Ana \<Gamma>"   and "\<Gamma> (Fun f X) = Fun g Y" shows "f = g"
split_list_first_prop_iff: fixes xs :: "'a list"   and P :: "'a \<Rightarrow> bool" shows "(\<exists>x\<in>set xs. P x) = (\<exists>ys x zs. xs = ys @ x # zs \<and> P x \<and> (\<forall>y\<in>set ys. \<not> P y))"
pseudonatural_equivalence_axioms: shows "pseudonatural_equivalence (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C (\<cdot>\<^sub>C) (\<star>\<^sub>C) \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C E.FG.map E.FG.cmp I\<^sub>C.map I\<^sub>C.cmp \<epsilon>\<^sub>0 \<epsilon>\<^sub>1"
Suc_pairwise: fixes P :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xs :: "'a stream" assumes "\<forall>i. P (xs !! i) (xs !! Suc i)" shows "PTA_Reachability.pairwise P xs"
countable_Image: fixes Y :: "'a set"   and X :: "('a \<times> 'b) set" assumes "\<And>y. y \<in> Y \<Longrightarrow> countable (X `` {y})"   and "countable Y" shows "countable (X `` Y)"
ffold_finsert_fremove: fixes z :: "nat"   and x :: "'e \<times> nat"   and A :: "('e \<times> nat) fset" shows "ffold tcf z (finsert x A) = tcf x (ffold tcf z (A |-| {|x|}))"
hyps_4: fixes fs' :: "(char list \<times> trm) list" assumes "fs [\<longmapsto>] fs'" shows "\<Gamma> \<turnstile> fs' [:] fTs"
succ_def: fixes less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and pos :: "'a \<Rightarrow> int"   and x :: "'a" assumes "class.spark_enum less_eq less pos" shows "spark_enum.succ pos x = spark_enum.val pos (pos x + 1)"
Py_nn: fixes x :: "'c" assumes "x \<in> space T" shows "0 \<le> Py x"
1: shows "reflexive (forest_components (forest_components f \<sqinter> w))"
a_max_c: fixes A :: "'a" assumes "A \<in> set As" shows "A = Max (atms_of (CA_of A))"
qtable_join_fixed: fixes n :: "nat"   and A :: "nat set"   and P :: "'a option list \<Rightarrow> bool"   and Q1 :: "'a option list \<Rightarrow> bool"   and X :: "'a option list set"   and B :: "nat set"   and Q2 :: "'a option list \<Rightarrow> bool"   and Y :: "'a option list set"   and b :: "bool"   and C :: "nat set" assumes "qtable n A P Q1 X"   and "qtable n B P Q2 Y"   and "\<not> b \<Longrightarrow> B \<subseteq> A"   and "C = A \<union> B"   and "\<And>x. \<lbrakk>wf_tuple n C x; P x\<rbrakk> \<Longrightarrow> P (restrict A x) \<and> P (restrict B x)" shows "qtable n C P (\<lambda>x. Q1 (restrict A x) \<and> (if b then Q2 (restrict B x) else \<not> Q2 (restrict B x))) (join X b Y)"
synth_inversion_1: fixes n :: "fresh_t"   and H :: "msg set" assumes "NonceF n \<in> synth H" obtains "NonceF n \<in> H"
F_mono: fixes x :: "real"   and y :: "real" assumes "a \<le> x"   and "x \<le> y"   and "y \<le> b" shows "F x \<le> F y"
Gromov_converging_at_boundary_imp_not_constant': fixes u :: "nat \<Rightarrow> 'a" assumes "Gromov_converging_at_boundary u" shows "\<not> (\<forall>m n. u m = u n)"
Units_l_inv: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c" assumes "Group.monoid G"   and "x \<in> Units G" shows "inv\<^bsub>G\<^esub> x \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>"
oalist_inv_rawD1: fixes xs :: "('a \<times> 'c) list" assumes "oalist_inv_raw xs" shows "(0::'c) \<notin> snd ` set xs"
domTrans: fixes a :: "'b \<Rightarrow> 'c option"   and b :: "'b \<Rightarrow> 'd option"   and c :: "'b \<Rightarrow> 'e option" assumes "dom a \<subseteq> dom b"   and "dom b \<inter> dom c = {}" shows "dom a \<inter> dom c = {}"
pr_conv_3_to_2_lm: fixes f :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" assumes "f \<in> PrimRec3" shows "pr_conv_3_to_2 f \<in> PrimRec2"
fset_mp: fixes A :: "'a fset"   and B :: "'a fset"   and c :: "'a" assumes "A |\<subseteq>| B"   and "c |\<in>| A" shows "c |\<in>| B"
hyps: fixes vs :: "'a list" assumes "v \<in> set vs" shows "v \<notin> f ` set (remdups_wrt_rev f xs vs)"
lexext_rev_cons_cons: fixes ys :: "'b list"   and xs :: "'b list"   and gt :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and y :: "'b"   and x :: "'b" assumes "length ys = length xs" shows "lexext_rev gt (y # ys) (x # xs) = (lexext_rev gt ys xs \<or> ys = xs \<and> gt y x)"
dfs_eq_reachable: fixes y :: "bool list"   and xs :: "bool list list" assumes "nfa_is_node A y"   and "list_all (nfa_is_node A) xs" shows "subset_memb y (dfs subset_empt xs) = (y \<in> reachable (set xs))"
Z2: shows "(r + 1) * (c + 1) = (q + 1) * (p + 1)"
set_injective: fixes X :: "V set"   and Y :: "V set" assumes "small X"   and "small Y" shows "(set X = set Y) = (X = Y)"
rv2_Inr_last: fixes s1 :: "'e \<Rightarrow> ('f, 'e, 'd) F1"   and s2 :: "'d \<Rightarrow> ('g, 'e, 'd) F2"   and kl :: "(bd_type_F + bd_type_F) list"   and xb :: "bd_type_F"   and x :: "'d" shows "\<exists>b2'. rv2 s1 s2 (kl @ [Inr xb]) x = Inr b2'"
2: shows "(f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e) * forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e) \<le> forest_components (f \<sqinter> - q \<squnion> (f \<sqinter> q)\<^sup>T \<squnion> e)"
IH_2: assumes "w \<Turnstile>\<^sub>n af_letter \<phi>2 x[Y]\<^sub>\<mu>" shows "(x ## w) \<Turnstile>\<^sub>n \<phi>2[Y]\<^sub>\<mu>"
error_free_Xcpt: fixes x :: "xcpt"   and s :: "st" shows "error_free (Some (Xcpt x), s)"
jb: fixes i :: "nat"   and j :: "nat" assumes "i < n"   and "j < n" shows "local.jb A i j"
MinVars_def: fixes min_lvar_not_in_bounds :: "('a, 'b) state \<Rightarrow> nat option"   and eq_idx_for_lvar :: "(nat \<times> linear_poly) list \<Rightarrow> nat \<Rightarrow> nat"   and min_rvar_incdec_eq :: "('a, 'b) Direction \<Rightarrow> ('a, 'b) state \<Rightarrow> nat \<times> linear_poly \<Rightarrow> 'a list + nat" shows "MinVars min_lvar_not_in_bounds eq_idx_for_lvar min_rvar_incdec_eq \<equiv> MinLVarNotInBounds min_lvar_not_in_bounds \<and> MinRVars eq_idx_for_lvar min_rvar_incdec_eq"
B: fixes self :: "\<AA> state \<times> \<AA> state \<Rightarrow> \<langle>\<langle>type\<^sub>P\<^sub>e\<^sub>r\<^sub>s\<^sub>o\<^sub>n\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "\<AA> state \<times> \<AA> state" shows "(\<lambda>_. Set{\<lambda>_. self \<tau>.salary} \<tau>) = (\<lambda>_. Set{self.salary} \<tau>)"
strict_trans2: fixes b :: "'c"   and a :: "'c"   and c :: "'c" assumes "b < a"   and "c \<le> b" shows "c < a"
vector_space_ow_axioms: shows "vector_space_ow U\<^sub>V\<^sub>S\<^sub>_\<^sub>2 (+\<^sub>V\<^sub>S\<^sub>_\<^sub>2) 0\<^sub>V\<^sub>S\<^sub>_\<^sub>2 (-\<^sub>V\<^sub>S\<^sub>_\<^sub>2) uminus\<^sub>V\<^sub>S\<^sub>_\<^sub>2 (*s\<^sub>o\<^sub>w\<^sub>_\<^sub>2)"
ereal_mult_indicator: fixes x :: "real"   and A :: "'b set"   and y :: "'b" shows "ereal (x * indicat_real A y) = ereal x * indicator A y"
pth: shows "pm_pth (dpm bdd) n = (v, x1a, x2a)"
as_needed_asses_submap_exec_ii: fixes v :: "'a"   and f :: "('a, 'b) fmap"   and g :: "('a, 'b) fmap" assumes "v \<in> fmdom' f"   and "f \<subseteq>\<^sub>f g" shows "fmlookup f v = fmlookup g v"
assms_3: fixes x :: "'x"   and h :: "'heap" assumes "x \<in> set xs" shows "pure (f x) h"
special_coset_pseudominimals: fixes X :: "'w set" assumes "dual_order.pseudominimal_in \<P> X" shows "\<exists>w s. w \<in> W \<and> s \<in> S \<and> X = w +o \<langle>S - {s}\<rangle>"
integrable_imp_measurable_weak: fixes S :: "'a set"   and f :: "'a \<Rightarrow> 'b" assumes "S \<in> sets lebesgue"   and "f integrable_on S" shows "f \<in> borel_measurable (lebesgue_on S)"
intros: fixes s :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and t :: "'t"   and x :: "'x"   and s' :: "('l, 't) locks \<times> (('t \<Rightarrow> ('x \<times> 'l \<Rightarrow>f nat) option) \<times> 'm) \<times> ('t \<Rightarrow> 'w wait_set_status option) \<times> 't set"   and x' :: "'x"   and ln' :: "'l \<Rightarrow>f nat"   and ta :: "('l, 't, 'x, 'm, 'w, 'o) thread_action" assumes "thr s t = \<lfloor>(x, no_wait_locks)\<rfloor>"   and "thr s' t = \<lfloor>(x', ln')\<rfloor>"   and "\<tau>move (x, shr s) ta (x', shr s')" shows "m\<tau>move s (t, ta) s'"
dedekind_2: fixes y :: "'a"   and x :: "'a"   and z :: "'a" shows "y * x \<sqinter> z \<le> (y \<sqinter> z * x\<^sup>T) * x"
hyps_3: fixes t :: "('fun, 'var) Term.term" assumes "t \<in> set T" shows "M' \<turnstile>\<^sub>c t"
minus_fpxs_def: fixes f :: "'b fpxs"   and g :: "'b fpxs" shows "f - g = f + - g"
dc_prop2: fixes x :: "('a \<times> 'a set) set" shows "mrel_c_lattice.d (x \<cdot> 1\<^sub>\<pi>) = mrel_c_lattice.d x"
fact_double: fixes n :: "nat" shows "fact (2 * n) = (2::'a) ^ (2 * n) * pochhammer ((1::'a) / (2::'a)) n * fact n"
deriv_ln_Gamma_real: fixes z :: "real" assumes "0 < z" shows "deriv ln_Gamma z = Digamma z"
injectivity: fixes x :: "'c poly"   and y :: "'c poly" assumes "map_poly to_fract x = map_poly to_fract y" shows "x = y"
runit'_in_hom_2: fixes f :: "'a" assumes "B.ide f" shows "\<guillemotleft>B.runit' f : f \<rightarrow>\<^sub>B f \<star>\<^sub>B src\<^sub>B f\<guillemotright>"
that: fixes d' :: "('test, 'atom) com"   and t' :: "'state" assumes "(d, t) \<rightarrow>c (d', t')"   and "s' \<approx> t'"   and "c' \<approx>s d'" shows "thesis"
insert_mono: fixes C :: "'c set"   and D :: "'c set"   and a :: "'c" assumes "C \<subseteq> D" shows "insert a C \<subseteq> insert a D"
assms_1: shows "f \<in> X \<rightarrow> Y"
broadcast_is_valid: fixes m :: "'a \<times> ('a, 'b) operation"   and i :: "nat" assumes "Broadcast m \<in> set (history i)" shows "\<exists>state. valid_rga_msg state m"
mtd: fixes r :: "('V Standard_Constant, 'b) labeled_graph \<times> ('V Standard_Constant, 'b) labeled_graph" assumes "maintained r G'" shows "maintained r G"
RP_sound: fixes Sts :: "('a literal multiset set \<times> 'a literal multiset set \<times> 'a literal multiset set) llist" assumes "chain (\<leadsto>) Sts"   and "{#} \<in> N_of_state (Liminf_state Sts) \<union> P_of_state (Liminf_state Sts) \<union> Q_of_state (Liminf_state Sts)" shows "\<not> satisfiable (grounding_of_clss (N_of_state (lhd Sts) \<union> P_of_state (lhd Sts) \<union> Q_of_state (lhd Sts)))"
HTset_r_def: fixes r :: "com \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool"   and \<pi> :: "char list \<Rightarrow> com option"   and \<Theta> :: "(((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool) \<times> com \<times> ((char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> (char list \<Rightarrow> int \<Rightarrow> int) \<Rightarrow> bool)) set" shows "HTset_r r \<pi> \<Theta> \<equiv> \<forall>(P, c, Q)\<in>\<Theta>. HT \<pi> (\<lambda>s. r c s \<and> P s) c Q"
induct: fixes x :: "'b list"   and C :: "'b list set"   and b :: "'b list"   and P :: "'b list \<Rightarrow> bool" assumes "x \<in> no_head_gen C b"   and "\<And>u. \<lbrakk>u \<in> C; u \<noteq> b\<rbrakk> \<Longrightarrow> P u"   and "\<And>u. \<lbrakk>u \<in> no_head_gen C b; P u\<rbrakk> \<Longrightarrow> P (u \<cdot> b)" shows "P x"
eq_dff_mult_right_trans: fixes x :: "'b fract"   and y :: "'b fract"   and z :: "'b fract"   and u :: "'b fract" assumes "x =dff y * z"   and "z =dff u" shows "x =dff y * u"
det_dim_zero: fixes A :: "'a mat" assumes "A \<in> carrier_mat 0 0" shows "Determinant.det A = (1::'a)"
delete_edges_empty: fixes G :: "'b graph" shows "delete_edges G {} = G"
distinct_disc_4: fixes "term" :: "'a local.term" assumes "is_Hcomp term" shows "\<not> is_Prim\<^sub>0 term"
resultant_swap: fixes f :: "'b poly"   and g :: "'b poly" shows "resultant f g = (- (1::'b)) ^ (degree f * degree g) * resultant g f"
1: fixes n :: "'node"   and v :: "'val"   and v' :: "'val" assumes "phis g (n, v) \<noteq> None"   and "phis g (n, v') \<noteq> None"   and "var g v = var g v'" shows "v = v'"
simps_2: fixes n :: "nat"   and f :: "'e \<Rightarrow> 'e" shows "f ^^ Suc n = f \<circ> f ^^ n"
edge_of_def: fixes e :: "'b set"   and S :: "'b set" shows "(e edge_of S) = (e face_of S \<and> aff_dim e = 1)"
adjuster_carrier: fixes w :: "'b vec"   and n :: "nat"   and us :: "'b vec list" assumes "w \<in> carrier_vec n"   and "set us \<subseteq> carrier_vec n"   and "distinct us" shows "adjuster n w us \<in> carrier_vec n"
assms_3: shows "\<forall>n. is_zero_row_upt_k n k A"
currentD_OUT': fixes f :: "'v \<times> 'v \<Rightarrow> ennreal"   and x :: "'v" assumes "current \<Gamma> f"   and "x \<notin> A \<Gamma>" shows "d_OUT f x = 0"
Disj_EH_2: fixes A :: "fm"   and A1 :: "fm"   and H :: "fm set"   and B :: "fm"   and Ba :: "fm" assumes "insert A (insert A1 H) \<turnstile> B"   and "insert Ba (insert A1 H) \<turnstile> B" shows "insert A1 (insert (A OR Ba) H) \<turnstile> B"
mkCone_cone: fixes a :: "'a"   and \<pi> :: "'b \<Rightarrow> 'a" assumes "D0.cone a \<pi>" shows "D0.mkCone \<pi> = \<pi>"
Pm_pref: shows "\<forall>m. Preorder (Pm m) \<and> Total (Pm m)"
count_def: fixes cs :: "('c \<Rightarrow> bool) list"   and a :: "'c"   and k :: "nat" shows "Degeneralization.count cs a k \<equiv> if k < length cs then if (cs ! k) a then Suc k mod length cs else k else if cs = [] then k else 0"
map_gpv'_comp: fixes f :: "'d \<Rightarrow> 'a"   and g :: "'e \<Rightarrow> 'b"   and h :: "'c \<Rightarrow> 'f"   and f' :: "'g \<Rightarrow> 'd"   and g' :: "'h \<Rightarrow> 'e"   and h' :: "'f \<Rightarrow> 'i"   and gpv :: "('g, 'h, 'i) gpv" shows "map_gpv' f g h (map_gpv' f' g' h' gpv) = map_gpv' (f \<circ> f') (g \<circ> g') (h' \<circ> h) gpv"
comp_arr_inv': fixes f :: "'a" assumes "local.iso f" shows "f \<cdot> local.inv f = cod f"
mp_in: shows "mix_pmf Up B W \<in> local.\<P>"
smalleraux3a: fixes a :: "('b, 'c) Combinators"   and b :: "('b, 'c) Combinators"   and l :: "'b set list" assumes "a \<noteq> DenyAll"   and "b \<noteq> DenyAll"   and "in_list b l"   and "in_list a l"   and "bothNet a \<noteq> bothNet b"   and "smaller a b l"   and "singleCombinators [a]"   and "singleCombinators [b]" shows "\<not> smaller b a l"
loop_imp_unsat: fixes x :: "nat"   and cnf :: "lit uprod set" assumes "(Pos x, Neg x) \<in> (imp_graph cnf)\<^sup>*"   and "(Neg x, Pos x) \<in> (imp_graph cnf)\<^sup>*" shows "\<not> satisfiable cnf"
set_zip_rightD: fixes x :: "'d"   and y :: "'e"   and xs :: "'d list"   and ys :: "'e list" assumes "(x, y) \<in> set (zip xs ys)" shows "y \<in> set ys"
step_in_range: fixes c :: "nat \<times> cell list \<times> cell list"   and A :: "(action \<times> nat) list" assumes "\<not> is_final (step0 c A)"   and "tm_wf0 A" shows "fst (step0 c A) \<le> length A div 2"
sorted_remove1: fixes less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and xs :: "'b list"   and a :: "'b" assumes "class.linorder less_eq less"   and "linorder.sorted less_eq xs" shows "linorder.sorted less_eq (remove1 a xs)"
step_plusI: fixes p :: "bool list"   and q :: "bool list"   and A :: "bool list \<times> ('b \<Rightarrow> bool list \<Rightarrow> bool list set) \<times> (bool list \<Rightarrow> bool)"   and a :: "'b" assumes "(p, q) \<in> step A a" shows "(p, q) \<in> step (RegExp2NA.plus A) a"
double_zero: 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 a :: "'b" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "(plus a a = zero) = (a = zero)"
\<theta>_simps_3: shows "trg \<chi>.the_\<theta> = src u"
assms_3: shows "D0 \<equiv> set bs \<times> (set gs \<union> set bs) -\<^sub>p set ps0"
interchange_rows_row_def: fixes A :: "(('b, 'c) vec, 'd) vec"   and a :: "'d"   and b :: "'d"   and i :: "'d" shows "interchange_rows_row A a b i = (\<chi>j. if i = a then A $ b $ j else if i = b then A $ a $ j else A $ i $ j)"
enc: fixes \<phi> :: "'c" assumes "\<phi> \<in> fmla" shows "\<langle>\<phi>\<rangle> \<in> num"
isTrivialPhi_def: fixes p :: "'g \<Rightarrow> 'node \<times> 'val \<Rightarrow> 'val list option"   and g :: "'g"   and v :: "'val"   and v' :: "'val" shows "uninst.isTrivialPhi p g v v' = (v' \<noteq> v \<and> (case uninst.phi p g v of None \<Rightarrow> False | Some vs \<Rightarrow> set vs = {v, v'} \<or> set vs = {v'}))"
sublist_def: fixes A :: "'d list"   and B :: "'d list" shows "sublist A B \<equiv> \<forall>x. List.member A x \<longrightarrow> List.member B x"
elims_1: fixes x :: "'a option"   and xa :: "'a option"   and y :: "bool" assumes "(x \<le> xa) = y" obtains xb :: "'a" and ya :: "'a" where "x = Some xb" and "xa = Some ya__" and "y = (xb \<le> ya__)"   | ya :: "'a option" where "x = None" and "xa = ya" and "y"   | uu_ :: "'a" where "x = Some uu_" and "xa = None" and "\<not> y"
open_INT: fixes "open" :: "'c set \<Rightarrow> bool"   and A :: "'d set"   and B :: "'d \<Rightarrow> 'c set" assumes "class.topological_space open"   and "finite A"   and "\<forall>x\<in>A. open (B x)" shows "open (\<Inter> (B ` A))"
updates_1: fixes i :: "nat" assumes "i \<le> m" shows "d fs'' i = (if i = i then (d fs (Suc i) * d fs (i - 1) + d\<mu> fs i (i - 1) * d\<mu> fs i (i - 1)) div d fs i else d fs i)"
set_of_add_inc_left: fixes A :: "'a interval"   and A' :: "'a interval"   and B :: "'a interval" assumes "set_of A \<subseteq> set_of A'" shows "set_of (A + B) \<subseteq> set_of (A' + B)"
pderiv_exp_gcd: shows "gcd p (pderiv p) = (\<Prod>(a, i)\<in>as. a ^ i)"
outer_loop_rel_wf: fixes src :: "nat" assumes "finite V" shows "wf (outer_loop_rel src)"
strict_implies_order: fixes a :: "'e list"   and b :: "'e list" assumes "strict_prefix a b" shows "prefix a b"
bind_assoc: fixes x :: "'d option"   and f :: "'d \<Rightarrow> 'c option"   and g :: "'c \<Rightarrow> 'b option" shows "Option.bind (Option.bind x f) g = Option.bind x (\<lambda>y. Option.bind (f y) g)"
op_in2c: shows "open (inside (path_image c2 \<union> path_image c))"
token_stays_in_final_states: fixes x :: "nat"   and n :: "nat"   and m :: "nat" assumes "token_run x n \<in> F" shows "token_run x (n + m) \<in> F"
simps_2: fixes \<NN> :: "symbol set"   and \<TT> :: "symbol set"   and \<RR> :: "(symbol \<times> symbol list) set"   and \<SS> :: "symbol"   and a :: "symbol list"   and d :: "nat \<times> symbol \<times> symbol list"   and D :: "(nat \<times> symbol \<times> symbol list) list"   and b :: "symbol list" assumes "CFG \<NN> \<TT> \<RR> \<SS>" shows "CFG.LeftDerivation \<NN> \<TT> \<RR> a (d # D) b = (\<exists>x. CFG.LeftDerives1 \<NN> \<TT> \<RR> a (fst d) (snd d) x \<and> CFG.LeftDerivation \<NN> \<TT> \<RR> x D b)"
facet_map: fixes C :: "'a set"   and z :: "'a set" assumes "domain.chamber C"   and "z \<lhd> C" shows "f ` z \<lhd> f ` C"
msg_zhops_simps_4: fixes d :: "nat"   and dip :: "nat" shows "msg_zhops (Newpkt d dip) = True"
r: shows "conv_radius (\<lambda>n. (- 1) ^ Suc n / of_nat n) = 1"
cfie: assumes "f\<inverse> \<in> {e, m, b, ov, s, f, d, m\<inverse>, b\<inverse>, ov\<inverse>, s\<inverse>, f\<inverse>, d\<inverse>}" shows "f\<inverse> O e = f\<inverse>"
FVS_def: fixes \<sigma> :: "('b, 'c, 'a) subst" shows "FVS \<sigma> = \<Union> (range (SFV \<sigma>))"
bdd_above_norm: fixes X :: "'a set" shows "bdd_above (norm ` X) = bounded X"
simps_4: fixes E :: "nat \<Rightarrow> 'a"   and F :: "'b \<Rightarrow> 'a list \<Rightarrow> 'a"   and G :: "'c \<Rightarrow> 'a list \<Rightarrow> bool"   and p :: "('b, 'c) fm" shows "\<lbrakk>E, F, G\<rbrakk> (\<^bold>\<forall> p) = (\<forall>x. \<lbrakk>E\<langle>0:x\<rangle>, F, G\<rbrakk> p)"
induct: 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 x1 :: "char list"   and x2 :: "char list list"   and Pa :: "char list \<Rightarrow> char list list \<Rightarrow> bool" assumes "Subobjs\<^sub>R P x1 x2"   and "\<And>C. is_class P C \<Longrightarrow> Pa C [C]"   and "\<And>C D Cs. \<lbrakk>P \<turnstile> C \<prec>\<^sub>R D; Subobjs\<^sub>R P D Cs; Pa D Cs\<rbrakk> \<Longrightarrow> Pa C (C # Cs)" shows "Pa x1 x2"
sup_red_f_in_red_liminf: fixes Ns :: "'f set llist" assumes "chain (\<rhd>) Ns" shows "Sup_llist (lmap Red_F Ns) \<subseteq> Red_F (Liminf_llist Ns)"
wset_final_ok_def: fixes ws :: "'d \<Rightarrow> 'e wait_set_status option"   and ts :: "'d \<Rightarrow> ('x \<times> 'f \<Rightarrow>f nat) option" shows "wset_final_ok ws ts = (\<forall>t\<in>dom ws. \<exists>x ln. ts t = \<lfloor>(x, ln)\<rfloor> \<and> \<not> final x)"
Deg_of_def: fixes w :: "floatarith" shows "Deg_of w = Num 180 * w / Pi"
cr: shows "change_respecting (cms, mem) (cms', mem') X g"
skip_elim: fixes mds :: "Mode \<Rightarrow> var\<^sub>C set"   and mem :: "var\<^sub>C \<Rightarrow> nat"   and c' :: "(var\<^sub>C, aexp\<^sub>C, bexp\<^sub>C) Stmt"   and mds' :: "Mode \<Rightarrow> var\<^sub>C set"   and mem' :: "var\<^sub>C \<Rightarrow> nat" assumes "eval_abv\<^sub>C \<langle>Skip, mds, mem\<rangle>\<^sub>C \<langle>c', mds', mem'\<rangle>\<^sub>C" shows "c' = Stop \<and> mds = mds' \<and> mem = mem'"
assms_2: fixes p :: "'p"   and w :: "('p, 'ba) label list"   and u' :: "'n"   and v :: "'n" assumes "eel = LOC (LCall p # w)"   and "(u, Call p, u') \<in> edges fg"   and "sh = [v, u']"   and "proc_of fg v = p"   and "mon_c fg ch = {}"   and "\<And>s. s \<in># ch \<Longrightarrow> \<exists>p u v. s = [entry fg p] \<and> (u, Spawn p, v) \<in> edges fg \<and> initialproc fg p"   and "\<And>P. {#[entry fg p]. p \<in># P#} \<subseteq># ch \<Longrightarrow> (v, mon_w fg w, P) \<in> S_cs fg (size P)" shows "Q"
differentiable_subset: fixes f :: "'b \<Rightarrow> 'c"   and x :: "'b"   and s :: "'b set"   and t :: "'b set" assumes "f differentiable at x within s"   and "t \<subseteq> s" shows "f differentiable at x within t"
res: shows "factorize_rat_poly_basic p = (c, qs)"
R'_1: shows "[v \<oplus> t]\<^sub>\<R> = region X I' r'"
nsets_self: fixes m :: "nat" shows "[{..<m}]\<^bsup>m\<^esup> = {{..<m}}"
f: fixes F :: "nat \<Rightarrow> 'a poly"   and n :: "nat \<Rightarrow> nat"   and \<delta> :: "nat \<Rightarrow> nat"   and f :: "nat \<Rightarrow> 'a"   and k :: "nat"   and \<beta> :: "nat \<Rightarrow> 'a"   and i :: "nat" assumes "primitive_remainder_sequence F n \<delta> f k \<beta>" shows "f i = lead_coeff (F i)"
sel_1: fixes even :: "bool"   and t :: "'a tree" shows "root (even_odd_mirror even t) = root t"