hcju/isabelleps huggingface dataset:
test_propE: fixes x :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "test x"   and "P (t x)" shows "P x"
Rep_uexpr_inject: fixes x :: "('d, 'c) uexpr"   and y :: "('d, 'c) uexpr" shows "(\<lbrakk>x\<rbrakk>\<^sub>e = \<lbrakk>y\<rbrakk>\<^sub>e) = (x = y)"
mset_le_addE: fixes xs :: "'c multiset"   and ys :: "'c multiset" assumes "xs \<subseteq># ys" obtains zs :: "'c multiset" where "ys = xs + zs"
intro: fixes less_eq0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less0 :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and idx0 :: "'a bound \<Rightarrow> 'a \<Rightarrow> nat"   and size0 :: "'a bound \<Rightarrow> nat"   and less_eq1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less1 :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and idx1 :: "'b bound \<Rightarrow> 'b \<Rightarrow> nat"   and size1 :: "'b bound \<Rightarrow> nat" assumes "index_locale less_eq0 less0 idx0 size0"   and "index_locale less_eq1 less1 idx1 size1" shows "prod_index less_eq0 less0 idx0 size0 less_eq1 less1 idx1 size1"
stable_no_blocking_onI: fixes X :: "'a set"   and ds :: "'b set" assumes "\<And>h X''. \<lbrakk>X'' = Ch h (X \<union> X''); X'' \<noteq> Ch h X; X'' \<subseteq> CD_on ds (X \<union> X'')\<rbrakk> \<Longrightarrow> False" shows "stable_no_blocking_on ds X"
3: shows "prv (cnj (PPf (encPf prf) \<langle>\<phi>R\<rangle>) (RR (encPf prf) \<langle>\<phi>R\<rangle>))"
pdevs_val_sum_le: fixes X :: "'b pdevs"   and d :: "nat"   and e :: "nat \<Rightarrow> real" assumes "degree X \<le> d" shows "pdevs_val e X = (\<Sum>i<d. e i *\<^sub>R pdevs_apply X i)"
worklists_vacant_to_def: fixes summary :: "'a \<Rightarrow> 'a \<Rightarrow> 'b antichain"   and results_in :: "'c \<Rightarrow> 'b \<Rightarrow> 'c"   and c :: "('a, 'c) configuration"   and t :: "'c" assumes "dataflow_topology summary results_in" shows "dataflow_topology.worklists_vacant_to summary results_in c t = (\<forall>loc1 loc2 s t'. s \<in>\<^sub>A graph.path_weight summary loc1 loc2 \<and> t' \<in>#\<^sub>z c_work c loc1 \<longrightarrow> \<not> results_in t' s \<le> t)"
d_bound: fixes fs :: "int vec list"   and ii :: "nat" assumes "LLL_invariant_weak fs"   and "g_bound fs"   and "ii < m" shows "d fs ii \<le> int (N ^ ii)"
linf_norm_vec_as_Greatest: fixes v :: "'b vec" shows "\<parallel>v\<parallel>\<^sub>\<infinity> = (GREATEST a. a \<in> abs ` set (list_of_vec v) \<union> {0::'b})"
uminus_r_inv_vec: fixes v :: "'a vec"   and n :: "nat" assumes "v \<in> carrier_vec n" shows "v + - v = 0\<^sub>v n"
ardens_lemma: fixes x :: "'a"   and z :: "'a"   and y :: "'a" assumes "\<not> ewp x"   and "z + x \<cdot> y = y" shows "x\<^sup>\<star> \<cdot> z = y"
a: shows "N[a\<turnstile>c>b] = Ax x c"
elims_3: fixes x :: "'a \<times> nat"   and xa :: "'a \<times> nat" assumes "\<not> happened_immediately_before x xa" obtains i :: "'a \<times> nat" and j :: "'a \<times> nat" where "x = i" and "xa = j" and "\<not> (is_valid_event_id i \<and> is_valid_event_id j \<and> (fst i = fst j \<and> Suc (snd i) = snd j \<or> is_reception i j))"
theI: fixes P :: "'c \<Rightarrow> bool"   and a :: "'c" assumes "P a"   and "\<And>x. P x \<Longrightarrow> x = a" shows "P (THE x. P x)"
LCons: fixes x :: "'c"   and xs :: "'c llist" assumes "x \<notin> lset xs"   and "ldistinct xs" shows "ldistinct (LCons x xs)"
sel_2: fixes x1 :: "'a"   and x2 :: "'a" shows "interval.upper (Interval x1 x2) = x2"
sgn_conic_sgn: fixes p :: "proj2" shows "sgn (conic_sgn p) = conic_sgn p"
refine: fixes Xi :: "'b list list"   and X :: "'c set"   and ivli :: "real list \<times> real list"   and ivl :: "'c set" assumes "(Xi, X) \<in> clw_rel appr_rel"   and "(ivli, ivl) \<in> lvivl_rel" shows "(RETURN (subset_spec_ivl_collc Xi ivli), subset_spec_coll $ X $ ivl) \<in> \<langle>bool_rel\<rangle>nres_rel"
aval''_correct: fixes s :: "char list \<Rightarrow> int"   and S :: "'av st option"   and a :: "aexp" assumes "s \<in> \<gamma>\<^sub>o S" shows "aval a s \<in> \<gamma> (aval'' a S)"
integrable_reflect_real: fixes f :: "real \<Rightarrow> 'a"   and b :: "real"   and a :: "real" shows "((\<lambda>x. f (- x)) integrable_on {- b..- a}) = (f integrable_on {a..b})"
fref_ncD: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'c \<Rightarrow> 'd"   and R :: "('a \<times> 'c) set"   and S :: "('b \<times> 'd) set" assumes "(f, g) \<in> R \<rightarrow>\<^sub>f S" shows "(f, g) \<in> R \<rightarrow> S"
sorted_wrt_map: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c list" shows "sorted_wrt R (map f xs) = sorted_wrt (\<lambda>x y. R (f x) (f y)) xs"
modes_respect_priv_def: fixes mdss\<^sub>C :: "(Mode \<Rightarrow> 'Var\<^sub>C set) list" shows "modes_respect_priv mdss\<^sub>C \<equiv> priv_is_asm_priv mdss\<^sub>C \<and> priv_is_guar_priv mdss\<^sub>C \<and> new_asms_only_for_priv mdss\<^sub>C \<and> new_asms_NoReadOrWrite_only mdss\<^sub>C"
While: fixes b :: "bexp"   and l :: "nat"   and c :: "com" assumes "max (sec b) l \<turnstile> c" shows "l \<turnstile> WHILE b DO c"
span_is_subspace: fixes S :: "'c set" assumes "S \<subseteq> carrier V" shows "subspace K (local.module.span S) V"
outside_union_restrict: fixes P :: "('c \<times> 'd) set"   and X :: "'c set" shows "P = Outside' X P \<union> P || X"
that: fixes q :: "('a \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b"   and t' :: "'a \<Rightarrow>\<^sub>0 nat" assumes "q \<in> hom_components p"   and "t' \<in> keys q"   and "t = monomial (poly_deg p - poly_deg q) x + t'" shows "thesis"
rep_eq: fixes x :: "('a, 'b) ffun"   and xa :: "('a, 'b) ffun" shows "pfun_of (x - xa) = pfun_of x - pfun_of xa"
once: shows "TSdet init h (as @ x # bs @ x # cs) (Suc (Suc (Suc (length as + length bs + i)))) = Partial_Cost_Model.Step (rTS h) (config\<^sub>p (rTS h) init (as @ x # bs @ x # take i cs)) (cs ! i)"
Bseq_subseq: fixes f :: "nat \<Rightarrow> 'a"   and g :: "nat \<Rightarrow> nat" assumes "Bseq f" shows "Bseq (\<lambda>x. f (g x))"
top_eq_inf_iff: fixes x :: "'a"   and y :: "'a" shows "(top = x \<sqinter> y) = (x = top \<and> y = top)"
l_new_character_data_get_tag_name_def: fixes get_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_new_character_data_get_tag_name get_tag_name_locs \<equiv> \<forall>h new_character_data_ptr h' r ptr'. h \<turnstile> new_character_data \<rightarrow>\<^sub>r new_character_data_ptr \<longrightarrow> h \<turnstile> new_character_data \<rightarrow>\<^sub>h h' \<longrightarrow> r \<in> get_tag_name_locs ptr' \<longrightarrow> r h h'"
lrule_sound: fixes SG :: "(('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list) list"   and C :: "('sf, 'sc, 'sz) formula list \<times> ('sf, 'sc, 'sz) formula list"   and i :: "nat"   and j :: "nat"   and L :: "lrule" assumes "lrule_ok SG C i j L"   and "i < length SG"   and "j < length (fst (SG ! i))"   and "sound (SG, C)" shows "sound (close (SG @ Lrule_result L j (SG ! i)) (SG ! i), C)"
winding_number_trivial: fixes z :: "complex"   and a :: "complex" assumes "z \<noteq> a" shows "winding_number (linepath a a) z = 0"
cases: fixes a :: "Regions.intv" assumes "Regions.isConst a" obtains uu_ :: "nat" where "a = Regions.intv.Const uu_"
nsum_in_VrTr: fixes v :: "'a \<Rightarrow> ant"   and n :: "nat"   and f :: "nat \<Rightarrow> 'a" assumes "valuation K v" shows "(\<forall>j\<le>n. f j \<in> carrier K) \<and> (\<forall>j\<le>n. 0 \<le> v (f j)) \<longrightarrow> \<Sigma>\<^sub>e K f n \<in> carrier (Vr K v)"
cSup_subset_mono: fixes Inf :: "'c set \<Rightarrow> 'c"   and Sup :: "'c set \<Rightarrow> 'c"   and inf :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and sup :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and A :: "'c set"   and B :: "'c set" assumes "class.conditionally_complete_lattice Inf Sup inf less_eq less sup"   and "A \<noteq> {}"   and "preorder.bdd_above less_eq B"   and "A \<subseteq> B" shows "less_eq (Sup A) (Sup B)"
More_2: fixes x :: "'a" assumes "x \<in> vars_ctxt (More f bef C aft)" shows "\<sigma> x = \<tau> x"
scale_zero_left: fixes x :: "'b" shows "0 *\<^sub>C x = (0::'b)"
meet_def': fixes x :: "'b"   and y :: "'b" shows "x \<sqinter> y = \<Sqinter>{x, y}"
distinct_1: shows "a\<^sub>c \<noteq> b\<^sub>c"
matrix_left_right_inverse: fixes A :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec"   and A' :: "(('a, 'b) Finite_Cartesian_Product.vec, 'b) Finite_Cartesian_Product.vec" shows "(A ** A' = mat (1::'a)) = (A' ** A = mat (1::'a))"
bij_imp_card_image: fixes f :: "'a \<Rightarrow> 'b"   and X :: "'a set" assumes "bij f" shows "card (f ` X) = card X"
span_explicit': fixes b :: "'b set" assumes "b \<subseteq> U\<^sub>V\<^sub>S" shows "local.span b = {x \<in> U\<^sub>V\<^sub>S. \<exists>f. x = (\<Sum>v\<in>{x \<in> U\<^sub>V\<^sub>S. f x \<noteq> (0::'a)}. f v *s v) \<and> finite {x \<in> U\<^sub>V\<^sub>S. f x \<noteq> (0::'a)} \<and> (\<forall>x\<in>U\<^sub>V\<^sub>S. f x \<noteq> (0::'a) \<longrightarrow> x \<in> b)}"
slength_ge_0: fixes xs :: "[:'r:]"   and n :: "int" assumes "slength\<cdot>xs = MkI\<cdot>n" shows "0 \<le> n"
list_assn_aux_simps_2: fixes P :: "'a \<Rightarrow> 'b \<Rightarrow> assn"   and l :: "'a list" shows "list_assn P l [] = \<up> (l = [])"
fun_relD1: fixes f :: "'c \<Rightarrow> 'd"   and f' :: "'e \<Rightarrow> 'f"   and Ra :: "('c \<times> 'e) set"   and Rr :: "('d \<times> 'f) set"   and x :: "'c"   and r :: "'d" assumes "(f, f') \<in> Ra \<rightarrow> Rr"   and "f x = r" shows "\<forall>x'. (x, x') \<in> Ra \<longrightarrow> (r, f' x') \<in> Rr"
monotone_eSuc: shows "monotone (\<le>) (\<le>) eSuc"
mirror: fixes p :: "fm"   and bbs :: "bool list"   and x :: "int"   and bs :: "int list" assumes "iszlfm p" shows "Ifm bbs (x # bs) (mirror p) = Ifm bbs (- x # bs) p"
IH: shows "(\<Sum>k = 1..<2 ^ n. f (2 ^ Discrete.log k)) = (\<Sum>k<n. 2 ^ k * f (2 ^ k))"
splus_term: fixes rep_nat_term :: "'a \<Rightarrow> (nat, nat) pp \<times> nat"   and splus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a" assumes "class.nat_term rep_nat_term splus" shows "rep_nat_term (splus x y) = pprod.splus (fst (rep_nat_term x)) (rep_nat_term y)"
assms_3: fixes p :: "nat" assumes "prime p" shows "f p \<noteq> 0"
assms: fixes x :: "'a"   and y :: "'b" assumes "Case_Labeling.SPLIT v (x, y)" shows "Case_Labeling.CTXT IC CT OC (ovalidNF (P x y) (B x y) Q)"
assms: shows "(\<Sum>i\<le>n. c i * a ^ i) = (0::'a)"
groupI: fixes G :: "('a, 'b) monoid_scheme" assumes "\<And>x y. \<lbrakk>x \<in> carrier G; y \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<in> carrier G"   and "\<one>\<^bsub>G\<^esub> \<in> carrier G"   and "\<And>x y z. \<lbrakk>x \<in> carrier G; y \<in> carrier G; z \<in> carrier G\<rbrakk> \<Longrightarrow> x \<otimes>\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> z = x \<otimes>\<^bsub>G\<^esub> (y \<otimes>\<^bsub>G\<^esub> z)"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<one>\<^bsub>G\<^esub> \<otimes>\<^bsub>G\<^esub> x = x"   and "\<And>x. x \<in> carrier G \<Longrightarrow> \<exists>y\<in>carrier G. y \<otimes>\<^bsub>G\<^esub> x = \<one>\<^bsub>G\<^esub>" shows "Group.group G"
simps_1: fixes e :: "char list \<Rightarrow> 'b option"   and x :: "char list"   and a :: "'b" shows "Env e\<lparr>x:a\<rparr> = (if x \<notin> dom e then Env (e(x \<mapsto> a)) else Malformed)"
subseq_Cons2: fixes xs :: "'a list"   and ys :: "'a list"   and x :: "'a" assumes "subseq xs ys" shows "subseq (x # xs) (x # ys)"
inj_sum_encode: fixes A :: "(nat + nat) set" shows "inj_on sum_encode A"
is_continuation_in_\<P>: fixes k :: "nat"   and u :: "nat"   and p :: "(symbol \<times> character list) list"   and ts :: "(symbol \<times> character list) list" assumes "is_continuation k u p ts" shows "p @ ts \<in> \<P> k (Suc u)"
adds_refl: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and zero :: "'c"   and a :: "'c" assumes "class.comm_monoid_add plus zero" shows "plus.adds plus a a"
start_ix: fixes d :: "nat"   and dm :: "nat" assumes "d < dm" shows "ix (start dm) d = 1"
elims_2: fixes x :: "nat \<times> nat list"   and xa :: "abc_inst list" assumes "abc_final x xa" obtains s :: "nat" and lm :: "nat list" and p :: "abc_inst list" where "x = (s, lm)" and "xa = p" and "s = length p"
suffix_tail: fixes s :: "nat \<Rightarrow> 'b" shows "s |\<^sub>s 1 = tail s"
path_unsnoc: fixes g :: "'g"   and ns :: "'node list" assumes "local.path g ns"   and "2 \<le> length ns" obtains "local.path g (butlast ns) \<and> last (butlast ns) \<in> set (predecessors g (last ns))"
final_schedule: shows "Liminf_llist (lmap fst Ns) = {}"
p_not_sys: fixes p :: "'d process_name" shows "(p \<noteq> sys) = (p = gc \<or> (\<exists>m. p = mutator m))"
isCont_def: fixes a :: "'d"   and f :: "'d \<Rightarrow> 'e" shows "isCont f a = f \<midarrow>a\<rightarrow> f a"
shortestPath_def: fixes g :: "'a"   and n :: "'node" shows "gE2.shortestPath g n \<equiv> LEAST l. \<exists>ns. length ns = l \<and> gE2.path2 g (Entry2 g) ns n"
size_Diff_submset: fixes M :: "'b multiset"   and M' :: "'b multiset" assumes "M \<subseteq># M'" shows "size (M' - M) = size M' - size M"
IH_2: fixes \<Gamma>' :: "nat \<Rightarrow> pdf_type" assumes "\<And>x. x \<in> free_vars e2 \<Longrightarrow> (case x of 0 \<Rightarrow> t1 | Suc x \<Rightarrow> \<Gamma> x) = \<Gamma>' x" shows "\<Gamma>' \<turnstile> e2 : t2"
Inf_upset_id: fixes Inf :: "'b set \<Rightarrow> 'b"   and Sup :: "'b set \<Rightarrow> 'b"   and inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bot :: "'b"   and top :: "'b"   and dual :: "'b \<Rightarrow> 'b" assumes "class.complete_lattice_with_dual Inf Sup inf less_eq less sup bot top dual" shows "Inf \<circ> ord.upset less_eq = id"
exponent_def: fixes f :: "Float.float" shows "Float.exponent f = snd (SOME p. f = 0 \<and> fst p = 0 \<and> snd p = 0 \<or> f \<noteq> 0 \<and> real_of_float f = real_of_int (fst p) * 2 powr real_of_int (snd p) \<and> odd (fst p))"
quit_iteration: fixes s :: "'a set"   and p :: "'b"   and s' :: "'a set"   and it :: "'c" shows "is_it s p s' it \<Longrightarrow>\<^sub>A is_set s p * true"
subs: shows "lhd Ns \<subseteq> Sup_llist Ns"
distr_id': fixes M :: "'c measure"   and N :: "'c measure" assumes "sets M = sets N" shows "distr M N (\<lambda>x. x) = M"
iv1: shows "size (tree_to_multiset (Node e1 a1 r ts1)) \<le> 2 ^ Suc r - 1"
all_prime_append: fixes ps :: "nat list"   and qs :: "nat list" shows "all_prime (ps @ qs) = (all_prime ps \<and> all_prime qs)"
If_eq: fixes y :: "'a"   and A :: "'a set"   and g :: "'a \<Rightarrow> 'c \<Rightarrow> 'b"   and h1 :: "'a \<Rightarrow> 'c"   and h2 :: "'a \<Rightarrow> 'c" assumes "y \<in> A"   and "finite A" shows "(\<Sum>x\<in>A. g x (if x = y then h1 x else h2 x)) = g y (h1 y) + (\<Sum>x\<in>A - {y}. g x (h2 x))"
word_bitwise_1_simps_1: shows "NOT 1 = - 2"
notin': shows "\<forall>j<length ts. let (uu_, uu_, uu_, sb\<^sub>j, uu_, uu_, uu_) = ts ! j in a \<notin> outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>j)"
t_halt_case_correctness: fixes ires :: "cell list"   and rs :: "nat"   and m :: "nat"   and n :: "nat" shows "let P = \<lambda>(st, l, r). st = 0; Q = \<lambda>(st, l, r). wcode_halt_case_inv st ires rs (l, r); f = steps0 (Suc 0, Bk # Bk \<up> m @ Oc # Bk # Bk # ires, Bk # Oc \<up> Suc rs @ Bk \<up> n) t_wcode_main in \<exists>n. P (f n) \<and> Q (f n)"
hyps_4: fixes ba :: "lam" assumes " P \<longrightarrow>\<^sub>1 ba" shows "\<exists>M3.  P' \<longrightarrow>\<^sub>1 M3 \<and>  ba \<longrightarrow>\<^sub>1 M3"
List_product_concat_map: fixes xs :: "'b list"   and ys :: "'c list" shows "List.product xs ys = concat (map (\<lambda>x. map (Pair x) ys) xs)"
approx_mult_subst: fixes u :: "'a star"   and v :: "'a star"   and x :: "'a star"   and y :: "'a star" assumes "u \<approx> v * x"   and "x \<approx> y"   and "v \<in> HFinite" shows "u \<approx> v * y"
m_closed: fixes x :: "'a"   and y :: "'a" assumes "x \<in> H"   and "y \<in> H" shows "x \<otimes> y \<in> H"
eval_in_hom: fixes t :: "'c local.term" assumes "Arr t" shows "\<guillemotleft>\<lbrace>t\<rbrace> : \<lbrace>Dom t\<rbrace> \<rightarrow> \<lbrace>Cod t\<rbrace>\<guillemotright>"
conj_zero_right: fixes x :: "'c" shows "x AND (0::'c) = (0::'c)"
is_cons_cluster_def: fixes quorum_of :: "'b \<Rightarrow> 'b set \<Rightarrow> bool"   and W :: "'b set"   and C :: "'b set" assumes "with_w quorum_of" shows "with_w.is_cons_cluster quorum_of W C \<equiv> C \<subseteq> W \<and> (\<forall>p\<in>C. \<exists>Q\<subseteq>C. quorum_of p Q) \<and> (\<forall>Q Q'. with_w.quorum_of_set quorum_of C Q \<and> with_w.quorum_of_set quorum_of C Q' \<longrightarrow> W \<inter> Q \<inter> Q' \<noteq> {})"
range_tmD: fixes x :: "real"   and e :: "nat \<Rightarrow> real"   and tm :: "taylor_model" assumes "x \<in>\<^sub>i range_tm e tm" shows "x \<in>\<^sub>i interval_of (insertion e (map_poly real_of_float (tm_poly tm))) + real_interval (tm_bound tm)"
cont_nn_integral_spmf: fixes f :: "'a \<Rightarrow> ennreal" shows "cont lub_spmf (ord_spmf (=)) Sup (\<le>) (\<lambda>p. integral\<^sup>N (measure_spmf p) f)"
exposed_face_of_Int_supporting_hyperplane_ge: fixes S :: "'b set"   and b :: "real"   and a :: "'b" assumes "convex S"   and "\<And>x. x \<in> S \<Longrightarrow> b \<le> a \<bullet> x" shows "S \<inter> {x. a \<bullet> x = b} exposed_face_of S"
case2: shows "direction linePR = 0 \<longrightarrow> line joining p to q \<parallel>  line joining p to r"
"1_1": shows "exec_plan (fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) s) as = exec_plan (fmrestrict_set (prob_dom (snapshot PROB (fmrestrict_set vs s))) s) as''"
Fvars_HPair: fixes t :: "tm"   and u :: "tm" shows "Fvars (HPair t u) = Fvars t \<union> Fvars u"
assms: fixes x :: "'a" assumes "x \<noteq> (0::'a)" shows "f x \<noteq> (0::'b)"
NewA: fixes G :: "prog"   and s0 :: "st"   and T :: "ty"   and s1 :: "state"   and e :: "expr"   and i' :: "val"   and s2 :: "state"   and a :: "loc"   and s3 :: "state" assumes "G\<turnstile>Norm s0 \<midarrow>init_comp_ty T\<rightarrow> s1"   and "G\<turnstile>s1 \<midarrow>e-\<succ>i'\<rightarrow> s2"   and "G\<turnstile>abupd (check_neg i') s2 \<midarrow>halloc Arr T (the_Intg i')\<succ>a\<rightarrow> s3" shows "G\<turnstile>Norm s0 \<midarrow>New T[e]-\<succ>Addr a\<rightarrow> s3"
cnf2bhv_greater_llength: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and c :: "'a"   and t :: "nat \<Rightarrow> cnf"   and n :: "nat"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b" assumes "dynamic_component.lActive active c t < n" shows "the_enat (llength (dynamic_component.proj tCMP active c (inf_llist t))) - 1 < dynamic_component.cnf2bhv tCMP active c t n"
yields_isomorphic_representation': obtains \<psi> :: "'a" where "\<guillemotleft>\<psi> : r\<^sub>0s\<^sub>1.p\<^sub>1 \<star> r\<^sub>0s\<^sub>1.p\<^sub>0\<^sup>* \<Rightarrow> r\<^sub>0\<^sup>* \<star> s\<^sub>1\<guillemotright>" and "local.iso \<psi>"
assms_2: shows "crdt_ops (ys @ [x] @ zs) deps"
sub1: shows "P \<turnstile> C'' \<prec>\<^sup>1 C'"
leqR_SubobjsR: fixes C :: "char list"   and D :: "char list"   and 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 wf_md :: "(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 \<Rightarrow> char list \<Rightarrow> char list \<times> ty list \<times> ty \<times> char list list \<times> expr \<Rightarrow> bool" assumes "(C, D) \<in> (subclsR P)\<^sup>*"   and "is_class P C"   and "wf_prog wf_md P" shows "\<exists>Cs. Subobjs\<^sub>R P C (Cs @ [D])"
induct: fixes P :: "'b up\<^sub>d \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> 'b \<Rightarrow> 'b tree234 \<Rightarrow> bool"   and a0 :: "'b up\<^sub>d"   and a1 :: "'b"   and a2 :: "'b tree234"   and a3 :: "'b"   and a4 :: "'b tree234" assumes "\<And>t1 a t2 b t3. P (T\<^sub>d t1) a t2 b t3"   and "\<And>t1 a t2 b t3 c t4. P (Up\<^sub>d t1) a \<langle>t2, b, t3\<rangle> c t4"   and "\<And>t1 a t2 b t3 c t4 d t5. P (Up\<^sub>d t1) a \<langle>t2, b, t3, c, t4\<rangle> d t5"   and "\<And>t1 a t2 b t3 c t4 d t5 e t6. P (Up\<^sub>d t1) a \<langle>t2, b, t3, c, t4, d, t5\<rangle> e t6"   and "\<And>v b d e. P (Up\<^sub>d v) b \<langle>\<rangle> d e" shows "P a0 a1 a2 a3 a4"
inv_mult_group: fixes G :: "('c, 'd) monoid_scheme"   and x :: "'c"   and y :: "'c" assumes "Group.group G"   and "x \<in> carrier G"   and "y \<in> carrier G" shows "inv\<^bsub>G\<^esub> (x \<otimes>\<^bsub>G\<^esub> y) = inv\<^bsub>G\<^esub> y \<otimes>\<^bsub>G\<^esub> inv\<^bsub>G\<^esub> x"
simps_2: fixes f :: "'a Bonk_Schramm_extension_unfolded \<Rightarrow> 'a Bonk_Schramm_extension_unfolded \<Rightarrow> 'a Bonk_Schramm_extension_unfolded \<Rightarrow> real"   and a :: "'a Bonk_Schramm_extension_unfolded"   and b :: "'a Bonk_Schramm_extension_unfolded" shows "extend_distance f (middle a b) = (\<lambda>y z. if y \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b) \<and> z \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b) then f (max2 Bonk_Schramm_extension_unfolded_wo y z) y z else if y \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b) \<and> z = middle a b then f (max2 Bonk_Schramm_extension_unfolded_wo a b) a b / 2 + (SUP w\<in>{z \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b). f z z z = 0}. f (max2 Bonk_Schramm_extension_unfolded_wo y w) y w - max (f (max2 Bonk_Schramm_extension_unfolded_wo a w) a w) (f (max2 Bonk_Schramm_extension_unfolded_wo b w) b w)) else if y = middle a b \<and> z \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b) then f (max2 Bonk_Schramm_extension_unfolded_wo a b) a b / 2 + (SUP w\<in>{z \<in> underS Bonk_Schramm_extension_unfolded_wo (middle a b). f z z z = 0}. f (max2 Bonk_Schramm_extension_unfolded_wo z w) z w - max (f (max2 Bonk_Schramm_extension_unfolded_wo a w) a w) (f (max2 Bonk_Schramm_extension_unfolded_wo b w) b w)) else if y = middle a b \<and> z = middle a b \<and> f a a a = 0 \<and> f b b b = 0 then 0 else 1)"
quasi_inverse_unique: fixes f :: "'d"   and g :: "'d"   and g' :: "'d" assumes "D.quasi_inverses f g"   and "D.quasi_inverses f g'" shows "D.isomorphic g g'"
generalise_output_preserves_label: fixes t :: "transition"   and r :: "nat"   and p :: "nat" shows "Label (generalise_output t r p) = Label t"
current_next_state: shows "\<forall>s execs. current (next_state s execs) = current s"
yx: shows "y = x \<or> y \<notin> Fvars \<phi>"
tensor_gate_unitary1: fixes m :: "nat"   and G1 :: "complex Matrix.mat"   and n :: "nat"   and G2 :: "complex Matrix.mat" assumes "gate m G1"   and "gate n G2" shows "G1 \<Otimes> G2\<^sup>\<dagger> * (G1 \<Otimes> G2) = 1\<^sub>m (dim_col G1 * dim_col G2)"
iFROM_cut_le: fixes n :: "nat"   and t :: "nat" shows "[n\<dots>] \<down>\<le> t = (if t < n then {} else [n\<dots>,t - n])"
sign_simps_32: fixes am :: "'c"   and bm :: "'c"   and ck :: "'c" shows "am * bm * ck = am * (bm * ck)"
isolated_verts_def: shows "G.isolated_verts \<equiv> {v \<in> verts G. out_arcs G v = {}}"
removeNones_vol_edges_not_f12': fixes y :: "nat"   and x :: "nat" assumes "y \<in> set (removeNones vol)" shows "(x, y) \<notin> \<E> f12"
and_mask_eq_iff_le_mask: fixes w :: "'a word"   and n :: "nat" shows "(w && mask n = w) = (w \<le> mask n)"
bezout_nat: fixes a :: "nat"   and b :: "nat" assumes "a \<noteq> 0" shows "\<exists>x y. a * x = b * y + gcd a b"
independent_mono: fixes A :: "'b set"   and B :: "'b set" assumes "local.independent A"   and "B \<subseteq> A" shows "local.independent B"
one_conv: fixes x :: "'a" shows "1' \<cdot> x ; 1 = 1' \<cdot> x ; x\<^sup>T"