hcju/isabelleps huggingface dataset:
nonzeros_trunc_pdevs_raw: fixes r :: "nat"   and x :: "nat \<Rightarrow> 'a" shows "{i. trunc_pdevs_raw r x i \<noteq> (0::'a)} \<subseteq> {i. x i \<noteq> (0::'a)}"
it_paths_strict_prefix: fixes "is" :: "nat list"   and t :: "'a rose_tree"   and is' :: "nat list" assumes "is \<in> it_paths t"   and "strict_prefix is' is" shows "is' \<in> it_paths t"
actions_asig_comp: fixes a :: "'d set \<times> 'd set \<times> 'd set"   and b :: "'d set \<times> 'd set \<times> 'd set" shows "actions (asig_comp a b) = actions a \<union> actions b"
rep_eq: fixes x :: "'a \<Rightarrow>\<^sub>C\<^sub>L 'b" shows "(*\<^sub>V) (sandwich x) = (\<lambda>B. x o\<^sub>C\<^sub>L B o\<^sub>C\<^sub>L x*)"
n'_def: shows "n' \<equiv> n(u := n u - {v})"
simps_10: fixes I :: "\<I>"   and \<psi> :: "'c MFOTL.formula" shows "gen_unique (formula.Next I \<psi>) = gen_unique \<psi>"
finite_complete_le_pair_complete: shows "finite_complete \<le> pair_complete"
assms_2: shows "AE x in M. h x \<noteq> 0"
times_divide_times_eq: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and w :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "times (divide x y) (divide z w) = divide (times x z) (times y w)"
del_UnEdge_node: fixes v :: "'a"   and u :: "'a"   and e :: "'b"   and u' :: "'a"   and G :: "('a, 'b, 'c) graph_scheme" shows "(v \<in> nodes (del_unEdge u e u' G)) = (v \<in> nodes G)"
NatTransComp_defs_1: fixes \<eta>1 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans"   and \<eta>2 :: "('g, 'h, 'i, 'j, 'k, 'l) NatTrans" shows "\<eta>1 \<bullet> \<eta>2 \<equiv> MakeNT (\<eta>1 \<bullet>1 \<eta>2)"
new_Addr_def: fixes s :: "state" shows "new_Addr s \<equiv> SOME v. (\<exists>a. v = Addr a \<and> heap s a = None) \<or> v = Null"
mult_funcset: shows "(\<otimes>) \<in> carrier G \<rightarrow> carrier G \<rightarrow> carrier G"
cts'_def_2: shows "ct_list_eq cts' (to_ct_list bs')"
adorn_def: fixes a :: "'d" shows "adorn a = pure (Pair a) \<diamondop> fetch"
unit_disc_fix_f_comp: fixes f1 :: "complex_homo \<Rightarrow> complex_homo"   and f2 :: "complex_homo \<Rightarrow> complex_homo" assumes "unit_disc_fix_f f1"   and "unit_disc_fix_f f2" shows "unit_disc_fix_f (f1 \<circ> f2)"
lowdim_subset_hyperplane: fixes S :: "'a set" assumes "dim S < DIM('a)" shows "\<exists>a. a \<noteq> (0::'a) \<and> span S \<subseteq> {x. a \<bullet> x = 0}"
thm_relation_negation_6: fixes v :: "i"   and p :: "\<o>" shows "[p \<^bold>\<noteq> p\<^sup>- in v]"
simps_2: fixes v :: "'c"   and l :: "'c Tree"   and r :: "'c Tree" shows "heapify (T v l r) = siftDown (T v (heapify l) (heapify r))"
MGT_Call: fixes \<Gamma> :: "'d \<Rightarrow> ('e, 'd, 'f) com option"   and \<Theta> :: "('e set \<times> 'd \<times> 'e set \<times> 'e set) set"   and F :: "'f set" shows "\<forall>p\<in>dom \<Gamma>. \<forall>Z. \<Gamma>,\<Theta>\<turnstile>\<^sub>t\<^bsub>/F\<^esub> {s. s = Z \<and> \<Gamma>\<turnstile> \<langle>Call p,Normal s\<rangle> \<Rightarrow>\<notin>{Stuck} \<union> Fault ` (- F) \<and> \<Gamma>\<turnstile>Call p \<down> Normal s} Call p {t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Normal t},{t. \<Gamma>\<turnstile> \<langle>Call p,Normal Z\<rangle> \<Rightarrow> Abrupt t}"
yoneda_lemma_for_hom_axioms: shows "yoneda_lemma_for_hom (\<cdot>) (\<cdot>\<^sub>S) \<phi> a a'"
proj_orig_basis_def: fixes n :: "nat"   and bs :: "('t \<Rightarrow>\<^sub>0 'c) list" shows "proj_orig_basis n bs = map (proj_poly_syz n) bs"
induct: fixes h :: "'a \<Rightarrow> 'b"   and bo :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and x1 :: "('a, 'b) rose_tree\<^sub>m"   and x2 :: "('a, 'b) rose_tree\<^sub>m"   and P :: "('a, 'b) rose_tree\<^sub>m \<Rightarrow> ('a, 'b) rose_tree\<^sub>m \<Rightarrow> bool" assumes "blinding_of_tree h bo x1 x2"   and "\<And>t1 t2. blinding_of_blindable (map_prod h (map (hash_tree h))) (rel_prod bo (list_all2 (\<lambda>x1 x2. blinding_of_tree h bo x1 x2 \<and> P x1 x2))) t1 t2 \<Longrightarrow> P (Tree\<^sub>m t1) (Tree\<^sub>m t2)" shows "P x1 x2"
A: shows "sinks g_I L High xs \<subseteq> {High}"
test_bit_eq_iff: fixes u :: "'b word"   and v :: "'b word" shows "((!!) u = (!!) v) = (u = v)"
simps: fixes phi :: "'c \<Rightarrow> 'd Filters.filter"   and x :: "'c"   and y :: "'d Filters.filter"   and z :: "'c"   and w :: "'d Filters.filter" assumes "triple phi" shows "triple.pairs_less_eq (x, y) (z, w) = (x \<le> z \<and> w \<le> y)"
d_s_subid: fixes x :: "('a \<times> 'a set) set" shows "(mrel_c_lattice.d x = x) = (x \<subseteq> 1\<^sub>\<sigma>)"
permutation_delete_bij_betw: fixes i :: "nat"   and n :: "nat"   and p :: "nat \<Rightarrow> nat" assumes "i \<in> {0..<Suc n}"   and "bij_betw p {0..<Suc n} {0..<Suc n}" shows "bij_betw (permutation_delete p i) {0..<n} {0..<n}"
add_rat: fixes b :: "int"   and d :: "int"   and a :: "int"   and c :: "int" assumes "b \<noteq> 0"   and "d \<noteq> 0" shows "Rat.Fract a b + Rat.Fract c d = Rat.Fract (a * d + c * b) (b * d)"
welldefined_subsets1: shows "all_list_constr subsets1 (length qs1)"
lt_higher: fixes v :: "'t"   and p :: "'t \<Rightarrow>\<^sub>0 'c" assumes "v \<prec>\<^sub>t lt p" shows "lt (higher p v) = lt p"
oa_contingent_5: fixes v :: "i" shows "[Contingent A! in v]"
globally_consistent_dom: fixes A :: "'Var \<Rightarrow> ('Val \<times> 'Val) option"   and mds :: "Mode \<Rightarrow> 'Var set"   and X :: "'Var set" assumes "globally_consistent A mds"   and "X \<subseteq> dom A" shows "globally_consistent (A |` X) mds"
assms_2: shows "typeof_addr h a = \<lfloor>Array_type T n\<rfloor>"
simps_3: shows "Falseif \<noteq> Trueif"
subset_iff_less_eq_V: fixes B :: "V set"   and A :: "V set" assumes "small B" shows "(A \<subseteq> B) = (ZFC_in_HOL.set A \<subseteq>\<^sub>\<circ> ZFC_in_HOL.set B \<and> small A)"
cases: fixes x :: "'b option" obtains "x = None"   | a :: "'b" where "x = Some a"
contI: fixes orda :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'd"   and luba :: "'c set \<Rightarrow> 'c"   and lubb :: "'d set \<Rightarrow> 'd"   and ordb :: "'d \<Rightarrow> 'd \<Rightarrow> bool" assumes "\<And>Y. \<lbrakk>Complete_Partial_Order.chain orda Y; Y \<noteq> {}\<rbrakk> \<Longrightarrow> f (luba Y) = lubb (f ` Y)" shows "cont luba orda lubb ordb f"
intros: fixes e :: "exp"   and \<Gamma> :: "'a"   and x :: "Vars.var"   and S :: "stack_elem list" assumes "isVal e" shows "boring_step (\<Gamma>, e, Upd x # S)"
islE: fixes x :: "'a + 'b" assumes "isl x" obtains l :: "'a" where "x = Inl l"
Heap_cases: fixes f :: "'a Heap"   and h :: "heap" obtains x :: "'a" and h' :: "heap" where "execute f h = Some (x, h')"   | "execute f h = None"
lookupDef_cases': fixes n :: "'node"   and g :: "'g"   and v :: "'var" assumes "n \<in> set (\<alpha>n g)" obtains "v \<in> defs g n" and "lookupDefNode g n v = n" and "defKind (lookupDef g n v) = SimpleDef"   | "v \<notin> defs g n" and "length (predecessors g n) \<noteq> 1" and "lookupDefNode g n v = n" and "defKind (lookupDef g n v) = PhiDef"   | m :: "'node" where "v \<notin> defs g n" and "predecessors g n = [m]" and "m \<in> set (\<alpha>n g)" and "lookupDef g n v = lookupDef g m v"
set_Cons_def: fixes A :: "'a set"   and XS :: "'a list set" shows "set_Cons A XS = {x # xs |x xs. x \<in> A \<and> xs \<in> XS}"
xxs: shows "x # xs = tuple_decode (pdec1 k) (pdec2 k)"
One_Lam: fixes x :: "name"   and t :: "lam"   and s :: "lam" assumes " Lam [x].t \<longrightarrow>\<^sub>1 s"   and "x \<sharp> s" shows "\<exists>t'. s = Lam [x].t' \<and>  t \<longrightarrow>\<^sub>1 t'"
nba_igbg_def: fixes A :: "('d, 'c) nba" shows "nba_igbg A \<equiv> graph_rec.extend (nba_g A) \<lparr>igbg_num_acc = 1, igbg_acc = \<lambda>p. if accepting A p then {0} else {}\<rparr>"
zip_same_conv_map: fixes xs :: "'g list" shows "zip xs xs = map (\<lambda>x. (x, x)) xs"
IH: fixes y :: "'a"   and b :: "'a"   and c :: "'a" assumes "R\<inverse>\<inverse> y x"   and "R\<^sup>*\<^sup>* y b"   and "R\<^sup>*\<^sup>* y c" shows "\<exists>d. R\<^sup>*\<^sup>* b d \<and> R\<^sup>*\<^sup>* c d"
unitor_coincidence_2: shows "\<r>\<^sub>1[\<I>\<^sub>1] = \<iota>\<^sub>1"
Rel_OR_simps_2: fixes x0 :: "'g"   and x1 :: "'a"   and w1 :: "'b" shows "(((x0, x1), Inr w1) \<in> Rel_OR) = ((x1, w1) \<in> Rel1 \<and> x0 \<in> valid_pub0)"
dvw_1: shows "decode_Var v \<noteq> decode_Var w"
simps_3: fixes f :: "'b \<Rightarrow> 'a pred"   and P :: "'b pred"   and xq :: "'b seq" shows "apply f (Join P xq) = Join (bind P f) (apply f xq)"
mgu: shows "Some \<sigma> = mgu (set_mset ` set (Map2.map2 add_mset As AAs))"
div_dvd_div: fixes a :: "'a"   and b :: "'a"   and c :: "'a" assumes "a dvd b"   and "a dvd c" shows "(b div a dvd c div a) = (b dvd c)"
closed_subsegmentI: fixes w :: "'b"   and x :: "'b"   and y :: "'b"   and z :: "'b"   and t :: "real" assumes "w \<in> {x--y}"   and "z \<in> {x--y}"   and "0 \<le> t"   and "t \<le> 1" shows "w + t *\<^sub>R (z - w) \<in> {x--y}"
inv_equality: fixes y :: "'a"   and x :: "'a" assumes "y \<otimes> x = \<one>"   and "x \<in> carrier \<G>"   and "y \<in> carrier \<G>" shows "inv x = y"
set_s_A: fixes acts :: "(nat \<times> nat list) list"   and qs :: "'c list"   and init :: "'c list"   and n :: "nat" assumes "BIT_Off acts qs init" shows "set (BIT_Off.s_A acts qs init n) = set init"
weight_less_iff: fixes w :: "'c \<times> nat \<Rightarrow> nat"   and w0 :: "nat"   and scf :: "'c \<times> nat \<Rightarrow> nat \<Rightarrow> nat"   and s :: "('c, 'd) Term.term"   and t :: "('c, 'd) Term.term" assumes "weight_fun.weight_less w w0 scf s t" shows "weight_fun.weight_le w w0 scf s t \<and> weight_fun.weight w w0 scf s < weight_fun.weight w w0 scf t"
substAbs_vsubstAbs_compose1: fixes A :: "('a, 'b, 'varSort, 'var, 'c) qAbs set"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and y :: "'var"   and y1 :: "'var"   and ys :: "'varSort" assumes "goodAbs A"   and "good Y"   and "y \<noteq> y1" shows "A $[y1 // y]_ys $[Y / y]_ys = A $[y1 // y]_ys"
trm_eqI: fixes a :: "'a"   and f :: "'a" assumes "ide a"   and "\<guillemotleft>f : a \<rightarrow> \<one>\<guillemotright>" shows "f = \<t>[a]"
Max_wrt_nonempty: assumes "alts \<noteq> {}"   and "finite alts" shows "Max_wrt R \<noteq> {}"
data_lookup: shows "data = \<Union> {set (rm_set_lookup rm k) |k. True}"
thetaFDW0_BisT: shows "thetaFDW0 \<subseteq> BisT"
neg_invariant_iff_compl_pos_invariant: fixes M :: "'a set" shows "negatively_invariant M = positively_invariant (X - M)"
cut_le_Min_not_empty: fixes I :: "'a set"   and t :: "'a" assumes "I \<noteq> {}"   and "iMin I \<le> t" shows "I \<down>\<le> t \<noteq> {}"
plus_current_simps: fixes f :: "'a \<times> 'a \<Rightarrow> ennreal"   and g :: "'a \<times> 'a \<Rightarrow> ennreal"   and e :: "'a \<times> 'a" shows "plus_current f g e = f e + g e"
nonneg: shows "\<forall>\<^sub>F x in F. 0 \<le> f x"
elims: fixes x :: "fml option"   and y :: "fml option" assumes "Noto x = y" obtains \<phi> :: "fml" where "x = Afml \<phi>" and "y = Afml (! \<phi>)"   | "x = undeff" and "y = undeff"
sourcenodes_is_n_Cons_butlast_targetnodes: fixes n :: "'node"   and as :: "'edge list"   and n' :: "'node" assumes "n -as\<rightarrow>* n'"   and "as \<noteq> []" shows "sourcenodes as = n # butlast (targetnodes as)"
power_diff_1_eq: fixes x :: "'b"   and n :: "nat" shows "x ^ n - (1::'b) = (x - (1::'b)) * sum ((^) x) {..<n}"
weak_reduction_coupled_simulation_versus_simulation_and_contrasimulation: fixes Rel :: "('b \<times> 'b) set"   and Cal :: "'b processCalculus" shows "weak_reduction_coupled_simulation Rel Cal = (weak_reduction_simulation Rel Cal \<and> weak_reduction_contrasimulation Rel Cal)"
simps_2: fixes f :: "'c \<Rightarrow> 'd \<Rightarrow> 'b"   and x1 :: "'c"   and x2 :: "'d" shows "(case (x1, x2) of (x, xa) \<Rightarrow> f x xa) = f x1 x2"
some_best_def: shows "local.some_best = (SOME x. x \<in> local.degenerate_lotteries \<and> x \<in> local.best)"
assms_4: shows "u \<turnstile>\<^bsub>v,n\<^esub> M"
nat_mirror_less_imp_less: fixes b :: "nat"   and l :: "nat"   and r :: "nat"   and a :: "nat" assumes "nat_mirror b l r < nat_mirror a l r" shows "a < b"
subgr: fixes H :: "'a set" assumes "H \<in> A" shows "subgroup H G"
tree_type: fixes t :: "'b bro"   and h :: "nat" assumes "t \<in> Bp h" shows "tree t \<in> B h \<union> B (Suc h)"
self_le_ge2_pow: fixes k :: "nat"   and m :: "nat" assumes "2 \<le> k" shows "m \<le> k ^ m"
fun_eq_on_trans: fixes f :: "'b \<Rightarrow> 'c"   and g :: "'b \<Rightarrow> 'c"   and A :: "'b set"   and h :: "'b \<Rightarrow> 'c" assumes "fun_eq_on f g A"   and "fun_eq_on g h A" shows "fun_eq_on f h A"
assms_1: shows "(f, g) \<in> A\<^sub>1 \<rightarrow> B\<^sub>1"
simps: fixes a :: "'b stream" shows "ssorted a = (\<exists>s. a = s \<and> shd s \<le> shd (stl s) \<and> ssorted (stl s))"
s_def: fixes t :: "real" shows "s t = (if t \<le> 0 then s0 else if t \<le> t_stop then p t else p_max)"
vdisjnt_mk_doubleton_image: fixes X :: "V"   and Y :: "V" shows "mk_doubleton_image X Y \<inter>\<^sub>\<circ> Y = []\<^sub>\<circ>"
intros: fixes p :: "perm"   and x :: "'a" shows "hull_relp (p \<bullet> x) x"
function_zero_eval: fixes a :: "'c" assumes "a \<in> S" shows "\<zero>\<^bsub>F\<^esub> a = \<zero>"
strict_prefixE': fixes xs :: "'c list"   and ys :: "'c list" assumes "strict_prefix xs ys" obtains z :: "'c" and zs :: "'c list" where "ys = xs @ z # zs"
tLength_Suc: fixes t :: "'a Trace"   and n :: "nat" assumes "tLength t = Suc n" shows "\<exists>s t'. t = t' \<leadsto> s \<and> tLength t' = n"
is_functor: shows "functor (\<cdot>\<^sub>J) (\<cdot>) Dom_f.map"
ae2ce_to_env_restr: fixes Aexp :: "exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and Aheap :: "(Vars.var \<times> exp) list \<Rightarrow> exp \<Rightarrow> Arity \<rightarrow> Vars.var \<Rightarrow> Arity\<^sub>\<bottom>"   and ae :: "Vars.var \<Rightarrow> Arity\<^sub>\<bottom>" assumes "NoCardinalityAnalysis Aexp Aheap" shows "ae2ce ae = (\<lambda>_. many) f|` edom ae"
upper_triangular_def: fixes A :: "(('b, 'c) vec, 'c) vec" shows "upper_triangular A = (\<forall>i j. j < i \<longrightarrow> A $ i $ j = (0::'b))"
intros: fixes a :: "nat"   and b :: "nat" shows "sum_of_2_squares_nat (a\<^sup>2 + b\<^sup>2)"
VInterI: fixes A :: "V"   and a :: "V" assumes "A \<noteq> 0"   and "\<And>x. x \<in>\<^sub>\<circ> A \<Longrightarrow> a \<in>\<^sub>\<circ> x" shows "a \<in>\<^sub>\<circ> \<Inter>\<^sub>\<circ> A"
isCont_sinc: fixes x :: "real" shows "isCont sinc x"
lookupDefE: fixes g :: "'g"   and n :: "'node"   and v :: "'var"   and v' :: "'var \<times> 'node \<times> Def" assumes "lookupDef g n v = v'"   and "n \<in> set (\<alpha>n g)" obtains "v \<in> defs g n" and "v' = (v, n, SimpleDef)"   | "v \<notin> defs g n" and "length (predecessors g n) \<noteq> 1" and "v' = (v, n, PhiDef)"   | m :: "'node" where "v \<notin> defs g n" and "predecessors g n = [m]" and "m \<in> set (\<alpha>n g)" and "v' = lookupDef g m v"
keys_single: fixes k :: "'c"   and v :: "'d" shows "keys (monomial v k) = (if v = (0::'d) then {} else {k})"
canceling_def: fixes a :: "bool \<times> 'e"   and b :: "bool \<times> 'e" shows "canceling a b = (snd a = snd b \<and> fst a \<noteq> fst b)"
strong_induct: fixes P :: "'a \<Rightarrow> trm \<Rightarrow> bool"   and z :: "'a"   and trm :: "trm" assumes "\<And>name z. P z (Var name)"   and "\<And>name trm z. \<lbrakk>name \<sharp> z; \<And>z. P z trm\<rbrakk> \<Longrightarrow> P z (Lam [name].trm)"   and "\<And>trm1 trm2 z. \<lbrakk>\<And>z. P z trm1; \<And>z. P z trm2\<rbrakk> \<Longrightarrow> P z (App trm1 trm2)" shows "P z trm"
pre: fixes F :: "'a \<Rightarrow> bool" assumes "\<turnstile> F" shows "|~ F"
start_point_no_predecessor: fixes x :: "'a" shows "x ; start_points x = 0"
analz_simp: fixes H :: "msg set" assumes "H \<subseteq> range Agent \<union> range Number \<union> range Nonce \<union> range Key \<union> range Hash \<union> range Passwd \<union> range Pri_AgrK \<union> range Pub_AgrK" shows "analz H = H"
in_homI: fixes f :: "'a \<times> 'a"   and a :: "'a \<times> 'a"   and b :: "'a \<times> 'a" assumes "VV.arr f"   and "VV.dom f = a"   and "VV.cod f = b" shows "VV.in_hom f a b"
bound_compare_defs_3: fixes lt :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and c :: "'a"   and b :: "'a option" shows "\<unlhd>\<^sub>u\<^sub>b lt c b \<equiv> case b of None \<Rightarrow> True | Some b' \<Rightarrow> lt c b' \<or> c = b'"
adjoint_pair_def: fixes f :: "'a"   and g :: "'a" shows "Cop.adjoint_pair f g \<equiv> \<exists>\<eta> \<epsilon>. adjunction_in_bicategory (\<cdot>) Cop.H Cop.\<a> Cop.\<i> Cop.src Cop.trg f g \<eta> \<epsilon>"
prob_space_imp_sigma_finite: fixes M :: "'c measure" assumes "prob_space M" shows "sigma_finite_measure M"
eqOnGL_def: fixes i :: "nat"   and i' :: "nat" shows "eqOnGL i i' \<equiv> scon {eqOnUOut i i' u ou |u ou. (u, ou) \<in> GL \<times> UNIV}"
not_in_interior_convex_hull_3_1: fixes a :: "complex"   and b :: "complex"   and c :: "complex" shows "a \<notin> interior (convex hull {a, b, c})"
vfsequence_of_vlist_vlist_of_vfsequence: shows "vfsequence_of_vlist (vlist_of_vfsequence xs) = xs"
Sigma_mono: fixes A :: "'e set"   and C :: "'e set"   and B :: "'e \<Rightarrow> 'f set"   and D :: "'e \<Rightarrow> 'f set" assumes "A \<subseteq> C"   and "\<And>x. x \<in> A \<Longrightarrow> B x \<subseteq> D x" shows "Sigma A B \<subseteq> Sigma C D"
power_less_imp_less_base: fixes a :: "'a"   and n :: "nat"   and b :: "'a" assumes "a ^ n < b ^ n"   and "(0::'a) \<le> b" shows "a < b"
elims_3: fixes x :: "'a ifex"   and xa :: "'a \<Rightarrow> bool" assumes "\<not> val_ifex x xa" obtains s :: "'a \<Rightarrow> bool" where "x = Falseif" and "xa = s__"   | n :: "'a" and t1 :: "'a ifex" and t2 :: "'a ifex" and s :: "'a \<Rightarrow> bool" where "x = IF n t1 t2" and "xa = s" and "\<not> (if s n then val_ifex t1 s else val_ifex t2 s)"
Q: shows "Q \<in> carrier_mat (dim_col A) (dim_col A)"
vmatch_eq0: fixes p :: "pat"   and v :: "value" shows "rel_option env_eq (vmatch p v) (smatch' p (value_to_sterm v))"
bounded_linear_imp_differentiable: fixes f :: "'c \<Rightarrow> 'd"   and net :: "'c filter" assumes "bounded_linear f" shows "f differentiable net"
all_final_except_mono: fixes A :: "'e set"   and B :: "'e set"   and ts :: "('f, 'e) locks \<times> (('e \<Rightarrow> ('x \<times> 'f \<Rightarrow>f nat) option) \<times> 'g) \<times> ('e \<Rightarrow> 'h wait_set_status option) \<times> 'e set" assumes "\<And>x. x \<in> A \<longrightarrow> x \<in> B" shows "all_final_except ts A \<longrightarrow> all_final_except ts B"
type_definition_Assertion: shows "type_definition Rep_Assertion Abs_Assertion assertion"
opposite_ocircline_cmat_def: fixes H :: "complex \<times> complex \<times> complex \<times> complex" shows "opposite_ocircline_cmat H = (- 1) *\<^sub>s\<^sub>m H"
that: assumes "0 < max 1 (max c c')"   and "\<And>n. n0 \<le> n \<Longrightarrow> norm (f n) \<le> max 1 (max c c') * norm (g n)" shows "thesis"
mor2ZFInj: fixes C :: "('d, 'e, 'f) LSCategory_scheme"   and x :: "'e"   and y :: "'e" assumes "LSCategory C"   and "x \<in> mor\<^bsub>C\<^esub>"   and "y \<in> mor\<^bsub>C\<^esub>"   and "m2z\<^bsub>C\<^esub>x = m2z\<^bsub>C\<^esub>y" shows "x = y"
flag_addpreRT: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and pre :: "nat set"   and ip :: "nat" assumes "dip \<in> kD rt" shows "flag (the (addpreRT rt dip pre)) ip = flag rt ip"
least_Suc: fixes A :: "nat set" assumes "A \<noteq> {}" shows "least (Suc ` A) = Suc (least A)"
that: fixes p :: "(('a \<times> 'b) \<times> 'c \<times> 'c) list" assumes "path PM p (initial PM)"   and "target p (initial PM) = FAIL" shows "thesis"
bit_or_iff: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and minus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and one :: "'c"   and divide :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and modulo :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and bit :: "'c \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and drop_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and take_bit :: "nat \<Rightarrow> 'c \<Rightarrow> 'c"   and "and" :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and or :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and xor :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and mask :: "nat \<Rightarrow> 'c"   and a :: "'c"   and b :: "'c"   and n :: "nat" assumes "class.semiring_bit_operations plus minus zero times one divide modulo bit push_bit drop_bit take_bit and or xor mask" shows "bit (or a b) n = (bit a n \<or> bit b n)"
Compositionality_If: fixes b :: "'exp"   and c1 :: "('exp, 'id) MWLsCom"   and c2 :: "('exp, 'id) MWLsCom"   and \<iota> :: "nat" assumes "\<forall>d. b \<equiv>\<^bsub>d\<^esub> b"   and "WHATWHERE_Secure [c1]"   and "WHATWHERE_Secure [c2]"   and "unique_PPc (if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi)" shows "WHATWHERE_Secure [if\<^bsub>\<iota>\<^esub> b then c1 else c2 fi]"
intros_2: fixes n :: "Arity"   and v1 :: "Value"   and v2 :: "Value" assumes "\<And>v. eq n (v1 \<down>Fn v) (v2 \<down>Fn v)" shows "eq (inc\<cdot>n) v1 v2"
cmp_simps_3: fixes f :: "'b"   and g :: "'b" assumes "B.ide f"   and "B.ide g"   and "src\<^sub>B f = trg\<^sub>B g" shows "trg\<^sub>C (\<Phi>\<^sub>H (f, g)) = trg\<^sub>C (H f)"
select_convs_1: fixes g_V :: "'a set"   and g_E :: "('a \<times> 'a) set"   and g_V0 :: "'a set"   and more :: "'b" shows "g_V \<lparr>g_V = g_V, g_E = g_E, g_V0 = g_V0, \<dots> = more\<rparr> = g_V"
f_rec: fixes x :: "nat" assumes "x\<^sub>1 \<le> x" shows "f x = g x + (\<Sum>i<k. as ! i * f ((ts ! i) x))"
nn_integral_sum: fixes P :: "'b set"   and f :: "'b \<Rightarrow> 'c \<Rightarrow> ennreal"   and M :: "'c measure" assumes "\<And>i. i \<in> P \<Longrightarrow> f i \<in> borel_measurable M" shows "\<integral>\<^sup>+ x. (\<Sum>i\<in>P. f i x) \<partial>M = (\<Sum>i\<in>P. integral\<^sup>N M (f i))"
swap_msgs_Recv_RecvMarker: fixes c :: "('a, 'b, 'c) configuration"   and ev :: "('a, 'b, 'c) event"   and d :: "('a, 'b, 'c) configuration"   and ev' :: "('a, 'b, 'c) event"   and e :: "('a, 'b, 'c) configuration"   and d' :: "('a, 'b, 'c) configuration"   and e' :: "('a, 'b, 'c) configuration"   and i :: "nat" assumes "c \<turnstile> ev \<mapsto> d"   and "d \<turnstile> ev' \<mapsto> e"   and "isRecv ev"   and "isRecvMarker ev'"   and "c \<turnstile> ev' \<mapsto> d'"   and "d' \<turnstile> ev \<mapsto> e'"   and "occurs_on ev \<noteq> occurs_on ev'" shows "msgs e i = msgs e' i"
funpow_mono: fixes f :: "'a \<Rightarrow> 'a"   and A :: "'a"   and B :: "'a"   and n :: "nat" assumes "mono f"   and "A \<le> B" shows "(f ^^ n) A \<le> (f ^^ n) B"
simps: fixes \<Sigma> :: "'a set set"   and \<phi> :: "'a ltl" shows "\<A>.\<A> \<Sigma> \<phi> = (\<A>.\<delta>\<^sub>\<A> \<Sigma>, \<A>.\<iota>\<^sub>\<A> \<phi>, \<A>.F\<^sub>\<A> \<Sigma> \<phi>)"
Sig_good_def: fixes \<Sigma> :: "((Class \<times> Method) \<times> nat) list" shows "Sig_good \<Sigma> = (\<forall>C m n. \<Sigma>\<down>(C, m) = Some n \<longrightarrow> MST\<down>(C, m) = Some (mkSPEC (Cachera n) emp) \<and> (\<exists>par code l0. mbody_is C m (par, code, l0)))"
proj_in_iff: fixes A :: "'d set"   and r :: "('d \<times> 'd) set"   and x :: "'d" assumes "equiv A r" shows "(Equiv_Relations.proj r x \<in> A // r) = (x \<in> A)"
numeral_le_one_iff: fixes one :: "'e"   and times :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and less_eq :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and less :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and plus :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and zero :: "'e"   and n :: "num" assumes "class.linordered_nonzero_semiring one times less_eq less plus zero" shows "less_eq (numeral.numeral one plus n) one = (n \<le> num.One)"
freshStarAtom: fixes xvec :: "name list"   and x :: "name" shows "xvec \<sharp>* x = x \<sharp> xvec"
takedrop: fixes n :: "nat"   and xs :: "'b list" shows "take n xs \<cdot> drop n xs = xs"