hcju/isabelleps huggingface dataset:
e_hd: fixes e :: "nat"   and x :: "nat"   and xs :: "nat list" assumes "Partial_Recursive.list_decode e = x # xs" shows "e_hd e = x"
lconjby_alternating_list_order2: fixes s :: "'a"   and t :: "'a"   and k :: "nat" assumes "s + s = (0::'a)"   and "t + t = (0::'a)" shows "lconjby (sum_list (alternating_list k s t)) (if gcd_nat.greater_eq k 2 then s else t) = sum_list (alternating_list (Suc (2 * k)) s t)"
vector_space_with_axioms_def: fixes plus\<^sub>V\<^sub>S :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and scale :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" shows "vector_space_with_axioms plus\<^sub>V\<^sub>S scale \<equiv> ((\<forall>a x y. scale a (plus\<^sub>V\<^sub>S x y) = plus\<^sub>V\<^sub>S (scale a x) (scale a y)) \<and> (\<forall>a b x. scale (a + b) x = plus\<^sub>V\<^sub>S (scale a x) (scale b x))) \<and> (\<forall>a b x. scale a (scale b x) = scale (a * b) x) \<and> (\<forall>x. scale (1::'d) x = x)"
UnaryRecOp_def: shows "UnaryRecOp = (\<lambda>g h. pr_conv_2_to_1 (PrimRecOp g (pr_conv_1_to_3 h)))"
payload_LtK: fixes X :: "ltkey"   and P :: "bool" assumes "LtK X \<in> payload" shows "P"
sup_state_conv2: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> 'b) list) list"   and s1 :: "ty list \<times> ty err list"   and s2 :: "ty list \<times> ty err list" shows "P \<turnstile> s1 \<le>\<^sub>i s2 = (P \<turnstile> fst s1 [\<le>] fst s2 \<and> P \<turnstile> snd s1 [\<le>\<^sub>\<top>] snd s2)"
simps_2: fixes R :: "'a set"   and x :: "nat" shows "min_arg R (Suc x) = (let j = min_arg R x in if S j \<inter> R = {} \<or> S (Suc x) \<inter> R \<noteq> {} \<and> cost R (Suc x) < cost R j then Suc x else j)"
get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t_ok: fixes h :: "(_) heap"   and ptr :: "(_) element_ptr"   and getter :: "('c, 'a, 'b, 'd, 'g option) RElement_scheme \<Rightarrow> 'm" assumes "ShadowRootClass.type_wf h"   and "ptr |\<in>| element_ptr_kinds h" shows "h \<turnstile> ok get_M\<^sub>E\<^sub>l\<^sub>e\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter"
time_closest_pair_rec_tm_simps_2: fixes xs :: "(int \<times> int) list" assumes "\<not> length xs \<le> 3" shows "time (closest_pair_rec_tm xs) = 1 + (let (xs\<^sub>L, xs\<^sub>R) = Time_Monad.val (split_at_tm (length xs div 2) xs); (ys\<^sub>L, p\<^sub>L) = Time_Monad.val (closest_pair_rec_tm xs\<^sub>L); (ys\<^sub>R, p\<^sub>R) = Time_Monad.val (closest_pair_rec_tm xs\<^sub>R); ys = Time_Monad.val (merge_tm snd ys\<^sub>L ys\<^sub>R) in time (length_tm xs) + time (split_at_tm (length xs div 2) xs) + time (closest_pair_rec_tm xs\<^sub>L) + time (closest_pair_rec_tm xs\<^sub>R) + time (merge_tm snd ys\<^sub>L ys\<^sub>R) + time (combine_tm p\<^sub>L p\<^sub>R (fst (hd xs\<^sub>R)) ys))"
strand_fv_subst_subset_if_subst_elim: fixes \<delta> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and v :: "'a"   and S :: "('b, 'a) strand_step list" assumes "subst_elim \<delta> v"   and "v \<in> fv\<^sub>s\<^sub>t S \<or> bvars\<^sub>s\<^sub>t S \<inter> (subst_domain \<delta> \<union> range_vars \<delta>) = {}" shows "v \<notin> fv\<^sub>s\<^sub>t (S \<cdot>\<^sub>s\<^sub>t \<delta>)"
family_Torder_def: fixes D :: "('c, 'd) Order_scheme" shows "fTo D = \<lparr>carrier = Algebra2.Chains D, rel = {Z \<in> Algebra2.Chains D \<times> Algebra2.Chains D. fst Z \<subseteq> snd Z}\<rparr>"
hyps_4: shows "fv t \<inter> insert v (subst_domain \<sigma>1) = {}"
ball_Un: fixes A :: "'b set"   and B :: "'b set"   and P :: "'b \<Rightarrow> bool" shows "(\<forall>x\<in>A \<union> B. P x) = ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>B. P x))"
oghoare_intros_23: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'a, 'c) com option"   and \<Theta> :: "'a \<Rightarrow> ('b, 'a, 'c) ann list option"   and F :: "'c set"   and P :: "'b set"   and g :: "'b set"   and a :: "('b, 'a, 'c) ann"   and c :: "('b, 'a, 'c) com"   and Q :: "'b set"   and A :: "'b set"   and f :: "'c"   and r :: "'b set" assumes "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> (P \<inter> g) a c Q, A"   and "P \<inter> - g \<noteq> {} \<Longrightarrow> f \<in> F" shows "\<Gamma>, \<Theta> \<tturnstile>\<^bsub>/F\<^esub> P (AnnRec r a) (Guard f g c) Q, A"
complex_add: fixes a :: "real"   and b :: "real"   and c :: "real"   and d :: "real" shows "Complex a b + Complex c d = Complex (a + c) (b + d)"
Ring_CongruenceI: fixes a :: "'a"   and b :: "'a" assumes "a - b \<in> I"   and "a \<in> R"   and "b \<in> R" shows "(a, b) \<in> Ring_Congruence"
part_eq: fixes x :: "i"   and y :: "i" assumes "M4" shows "P x y = (PP x y \<or> x = y)"
share_read_only_mono_in: fixes a :: "nat"   and sb :: "'b memref list"   and \<S> :: "nat \<Rightarrow> bool option"   and \<S>' :: "nat \<Rightarrow> bool option" assumes "a \<in> read_only (share sb \<S>)"   and "read_only \<S> \<subseteq> read_only \<S>'" shows "a \<in> read_only (share sb \<S>')"
distinct_1: fixes a :: "'g"   and b' :: "'h" shows "Inl a \<noteq> Inr b'"
gbm_delete_edge_impl: shows "graph_delete_edge gbm_\<alpha> gbm_invar gbm_delete_edge"
is_red_alt: fixes F :: "('t \<Rightarrow>\<^sub>0 'c) set"   and a :: "'t \<Rightarrow>\<^sub>0 'c" shows "is_red F a = (\<exists>b. red F a b)"
ParallelCom_def: fixes Ps :: "('b com \<times> 'b set \<times> ('b \<times> 'b) set \<times> ('b \<times> 'b) set \<times> 'b set) list" shows "ParallelCom Ps \<equiv> map (Some \<circ> fst) Ps"
vimageI2: fixes f :: "'b \<Rightarrow> 'a"   and a :: "'b"   and A :: "'a set" assumes "f a \<in> A" shows "a \<in> f -` A"
iso_hcomp: fixes V :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and src :: "'c \<Rightarrow> 'c"   and trg :: "'c \<Rightarrow> 'c"   and \<mu> :: "'c"   and \<nu> :: "'c" assumes "horizontal_composition V H src trg"   and "category.iso V \<mu>"   and "category.iso V \<nu>"   and "src \<nu> = trg \<mu>" shows "category.iso V (H \<nu> \<mu>)"
GB_alt_3_finite: fixes G :: "(('x \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'b) set" assumes "finite G" shows "pp_pm.punit.is_Groebner_basis G = (\<forall>f\<in>MPoly_Type_Class.punit.pmdl G. f \<noteq> 0 \<longrightarrow> (\<exists>g\<in>G. g \<noteq> 0 \<and> MPoly_Type_Class.punit.adds_term (pp_pm.lpp g) (pp_pm.lpp f)))"
hyps: fixes u :: "'a"   and v :: "'a" assumes "awalk u es v"   and "\<And>w1. w1 \<in> verts G \<Longrightarrow> P w1 [] w1"   and "\<And>e w1 w2 es. \<lbrakk>e \<in> arcs G; arc_to_ends G e = (w1, w2); P w2 es v\<rbrakk> \<Longrightarrow> P w1 (e # es) v" shows "P u es v"
is_subprob_density_def: fixes M :: "'c measure"   and f :: "'c \<Rightarrow> ennreal" shows "is_subprob_density M f \<equiv> f \<in> borel_measurable M \<and> space M \<noteq> {} \<and> (\<forall>x\<in>space M. 0 \<le> f x) \<and> integral\<^sup>N M f \<le> 1"
length_bound2: fixes p :: "nat"   and q :: "nat"   and m :: "bit list" shows "8 \<le> length (bvxor (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))) (MGF (sha1 (generate_M' (sha1 m) salt)) (length (generate_DB (generate_PS (length (nat_to_bv (p * q)) - Suc 0) (length (sha1 (generate_M' (sha1 m) salt))))))))"
finite_psupset_def: fixes S :: "'b set" shows "finite_psupset S \<equiv> {(Q', Q). Q \<subset> Q' \<and> Q' \<subseteq> S}"
rank_meldUniq: fixes bq1 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list"   and bq2 :: "('a, 'b) SkewBinomialHeapStruc.SkewBinomialTree list" assumes "SkewBinomialHeapStruc.rank_invar bq1"   and "SkewBinomialHeapStruc.rank_invar bq2" shows "SkewBinomialHeapStruc.rank_invar (SkewBinomialHeapStruc.meldUniq bq1 bq2)"
HARoot_SAs: fixes HA :: "('d, 'e, 'f) hierauto" shows "HARoot HA \<in> SAs HA"
A_eq_xs: shows "Gauss_Jordan_iarrays A = IArray xs"
that: fixes q1 :: "enat"   and q2 :: "enat" assumes "q1 + q2 \<le> q"   and "interaction_any_bounded_by \<A>1 q1"   and "\<And>cipher \<sigma>. interaction_any_bounded_by (\<A>2 cipher \<sigma>) q2" shows "thesis"
labeled_sat_ineq_dbproj: fixes M :: "('fun, 'var) Term.term set"   and X :: "'var list"   and t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term"   and Di :: "('a \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and D :: "('a \<times> ('fun, 'var) Term.term \<times> ('fun, 'var) Term.term) list"   and \<I> :: "'var \<Rightarrow> ('fun, 'var) Term.term"   and i :: "'a" assumes "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) D)\<rbrakk>\<^sub>d \<I>" shows "\<lbrakk>M; map (\<lambda>d. \<forall>X\<langle>\<or>\<noteq>: [(pair (t, s), pair (snd d))]\<rangle>\<^sub>s\<^sub>t) (filter (\<lambda>d. d \<notin> set Di) (dbproj i D))\<rbrakk>\<^sub>d \<I>"
field_def: fixes R :: "('a, 'b) ring_scheme" shows "field R \<equiv> domain R \<and> field_axioms R"
ys_def: shows "ys \<equiv> map (\<lambda>(x, y). x * prod_list (remove1 y (map snd xs))) xs"
sqrt_irrat_def: shows "sqrt_irrat = {q. \<nexists>p. p * p = rat_of_nat q}"
un_empty_absorb1: fixes i :: "nat_int" shows "i \<squnion> bot = i"
pref_profile_wf_axioms: shows "pref_profile_wf agents alts R"
hom_carrier: fixes h :: "'b \<Rightarrow> 'c"   and G :: "('b, 'd) monoid_scheme"   and H :: "('c, 'e) monoid_scheme" assumes "h \<in> hom G H" shows "h ` carrier G \<subseteq> carrier H"
coprime_r_def: fixes ps :: "real poly list" shows "coprime_r ps = pderiv (prod_list ps) * [:real_of_int (- crb (prod_list ps)), 1:] * [:real_of_int (crb (prod_list ps)), 1:]"
vars_head_subseteq: fixes s :: "('d, 'c) tm" shows "vars_hd (head s) \<subseteq> vars s"
cyclic_on_perm_restrict: fixes f :: "'b \<Rightarrow> 'b"   and S :: "'b set" shows "cyclic_on (perm_restrict f S) S = cyclic_on f S"
intros_2: fixes m1 :: "'a"   and i :: "nat"   and m2 :: "'a" assumes "Deliver m1 \<sqsubset>\<^sup>i Broadcast m2" shows "hb m1 m2"
is_imgu_def: fixes \<sigma> :: "'a \<Rightarrow> ('b, 'a) Term.term"   and E :: "(('b, 'a) Term.term \<times> ('b, 'a) Term.term) set" shows "is_imgu \<sigma> E = (\<sigma> \<in> unifiers E \<and> (\<forall>\<tau>\<in>unifiers E. \<tau> = \<sigma> \<circ>\<^sub>s \<tau>))"
order_le_less_subst1: fixes a :: "'a"   and f :: "'b \<Rightarrow> 'a"   and b :: "'b"   and c :: "'b" assumes "a \<le> f b"   and "b < c"   and "\<And>x y. x < y \<Longrightarrow> f x < f y" shows "a < f c"
prems: shows "Lyndon_mono (xs \<cdot> [x])"
IH_4: fixes x :: "nat"   and xa :: "nat"   and xaa :: "'a"   and xb :: "'a" assumes "x = dim_row A"   and "xa = dim_col A"   and "i < x \<and> j < xa"   and "xaa = A $$ (i, j)"   and "xaa \<noteq> (0::'a)"   and "xaa \<noteq> (1::'a)"   and "xb = inverse xaa"   and "multrow i xb A \<in> carrier_mat nr nc"   and "multrow i xb B \<in> carrier_mat nr nc2" shows "gauss_jordan_main (mat\<^sub>h (multrow i xb A)) (mat\<^sub>h (multrow i xb B)) i j = map_prod mat\<^sub>h mat\<^sub>h (gauss_jordan_main (multrow i xb A) (multrow i xb B) i j)"
invpst_baliL: fixes l :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and r :: "(('c \<times> 'd) \<times> tcolor \<times> 'c \<times> 'd) tree"   and a :: "'c \<times> 'd" assumes "invpst l"   and "invpst r" shows "invpst (baliL l a r)"
less_multiset_ext\<^sub>D\<^sub>M_def: fixes R :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and M :: "'b multiset"   and N :: "'b multiset" shows "less_multiset_ext\<^sub>D\<^sub>M R M N = (\<exists>X Y. X \<noteq> {#} \<and> X \<subseteq># N \<and> M = N - X + Y \<and> (\<forall>k. k \<in># Y \<longrightarrow> (\<exists>a. a \<in># X \<and> R k a)))"
pt_pi_rev: fixes pi :: "('b \<times> 'b) list"   and x :: "'a" assumes "pt TYPE('a) TYPE('b)"   and "at TYPE('b)" shows "pi \<bullet> rev pi \<bullet> x = x"
poly_rel_inj: fixes x :: "'i list"   and y :: "'a poly"   and z :: "'a poly" assumes "poly_rel x y"   and "poly_rel x z" shows "y = z"
bounded_iff: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "(sup (b::'b) (c::'b) \<le> (a::'b)) = (b \<le> a \<and> c \<le> a)"
semiring_simprules_5: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c"   and z :: "'c" assumes "semiring R"   and "x \<in> carrier R"   and "y \<in> carrier R"   and "z \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> y \<oplus>\<^bsub>R\<^esub> z = x \<oplus>\<^bsub>R\<^esub> (y \<oplus>\<^bsub>R\<^esub> z)"
run_scons_elim: fixes execute :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and enabled :: "'a \<Rightarrow> 'b \<Rightarrow> bool"   and a :: "'a"   and r :: "'a stream"   and p :: "'b" assumes "transition_system.run execute enabled (a ## r) p" obtains "enabled a p" and "transition_system.run execute enabled r (execute a p)"
min_dep_imp_ex1_supset_circuit: fixes x :: "'a"   and X :: "'a set" assumes "x \<in> \<E>"   and "indep_in \<E> X"   and "\<not> indep_in \<E> (insert x X)" shows "\<exists>!C. \<E>.circuit C \<and> C \<subseteq> insert x X"
rest_2: shows "\<forall>fe'\<in>set ft. ofe_prio fe < ofe_prio fe' \<longrightarrow> \<not> \<gamma> (ofe_fields fe') p"
infchain_outruns_all: fixes \<tau> :: "'a set \<Rightarrow> 'a set" assumes "finite UNIV"   and "\<forall>i. (\<tau> ^^ i) {} \<subset> (\<tau> ^^ (i + 1)) {}" shows "\<forall>j. \<exists>i. j < card ((\<tau> ^^ i) {})"
mopup_b_def: shows "mopup_b \<equiv> [(R, 2), (R, 1), (L, 5), (W0, 3), (R, 4), (W0, 3), (R, 2), (W0, 3), (L, 5), (L, 6), (R, 0), (L, 6)]"
assms: shows "A\<^sub>F \<sharp>* \<Psi>"
lt_monomial: fixes c :: "'b"   and t :: "'x \<Rightarrow>\<^sub>0 nat" assumes "c \<noteq> (0::'b)" shows "lpp (monomial c t) = t"
Bernstein_changes_def: fixes p :: "nat"   and c :: "real"   and d :: "real"   and P :: "real poly" shows "Bernstein_changes p c d P = int (nat (changes (Bernstein_coeffs p c d P)))"
min_maxsimpchainD_min_betw: fixes X :: "'b set set"   and x :: "'b set"   and xs :: "'b set list"   and y :: "'b set"   and ys :: "'b set list" assumes "SimplicialComplex X"   and "SimplicialComplex.min_maxsimpchain X (x # xs @ [y])"   and "SimplicialComplex.maxsimpchain X (x # ys @ [y])" shows "order.greater_eq (length ys) (length xs)"
del_point_def: fixes bl :: "'a set"   and p :: "'a" shows "add_block_sys.del_point bl p \<equiv> \<V> \<union> bl - {p}"
simps_8: fixes \<phi> :: "formula"   and I :: "\<I>"   and \<psi> :: "formula" shows "convert_multiway (Since \<phi> I \<psi>) = Since (convert_multiway \<phi>) I (convert_multiway \<psi>)"
shows_list_append: fixes shows_prec :: "nat \<Rightarrow> 'a \<Rightarrow> char list \<Rightarrow> char list"   and shows_list :: "'a list \<Rightarrow> char list \<Rightarrow> char list"   and xs :: "'a list"   and r :: "char list"   and s :: "char list" assumes "class.show shows_prec shows_list" shows "shows_list xs (r @ s) = shows_list xs r @ s"
NQ_def: shows "NQ = Neg ''Q'' [Fun ''d'' []]"
ipurge_ref_aux_foldr_eq: fixes I :: "('f \<times> 'f) set"   and D :: "'e \<Rightarrow> 'f"   and U :: "'f set"   and xs :: "'e list"   and X :: "'e set"   and P :: "'e \<Rightarrow> bool" shows "ipurge_ref_aux I D U xs X = ipurge_ref_aux_foldr I D P U xs X"
verticesFrom_pres_nodes1: fixes v :: "nat"   and f :: "face" assumes "v \<in> \<V> f" shows "\<V> f = set (verticesFrom f v)"
rel_gpv'_def: fixes A :: "'d \<Rightarrow> 'g \<Rightarrow> bool" shows "rel_gpv' A = rel_gpv A (=)"
psi_u_1: shows "psi_u \<in> \<R>\<^sup>2"
less_aexp_aux_trans: fixes x :: "'b aexp"   and y :: "'b aexp"   and z :: "'b aexp" assumes "AExp_Lexorder.less_aexp_aux x y"   and "AExp_Lexorder.less_aexp_aux y z" shows "AExp_Lexorder.less_aexp_aux x z"
sdom_dom: fixes n1 :: "nat"   and n2 :: "nat" assumes "strict_dominate n1 n2" shows "dominate n1 n2"
map_snd_mk_snd: fixes k :: "'a"   and l :: "'b list" shows "map snd (map (\<lambda>x. (x, k)) l) = replicate (length l) k"
relcomppI: fixes r :: "'j \<Rightarrow> 'k \<Rightarrow> bool"   and a :: "'j"   and b :: "'k"   and s :: "'k \<Rightarrow> 'l \<Rightarrow> bool"   and c :: "'l" assumes "r a b"   and "s b c" shows "(r OO s) a c"
res_e: shows "ord_resolve CAs (D + negs (mset As)) AAs As (\<Sum>\<^sub># (mset Cs) + D)"
insert_absorb2: fixes x :: "'d"   and A :: "'d set" shows "insert x (insert x A) = insert x A"
ga_gex_def: fixes ga :: "nat \<times> 'c \<times> 'a \<times> 'c"   and gc :: "('c, 'ls, 'gs) pid_global_config" shows "ga_gex ga gc \<equiv> let (pid, c, a, c') = ga; fs = fs_of_pid gc pid; (ls', gs') = ex fs a; gc' = \<lparr>pid_global_config.processes = (pid_global_config.processes gc)[pid := \<lparr>command = c', state = ls'\<rparr>], state = gs'\<rparr> in gc'"
bddh_exists: fixes B :: "'a bdd" shows "\<exists>n. bddh n B"
param_concat: fixes R :: "('g \<times> 'h) set" shows "(concat, concat) \<in> \<langle>\<langle>R\<rangle>list_rel\<rangle>list_rel \<rightarrow> \<langle>R\<rangle>list_rel"
negligible_cmult: fixes c :: "real"   and f :: "nat \<Rightarrow> real" shows "Negligible.negligible (\<lambda>\<eta>. c * f \<eta>) = (Negligible.negligible f \<or> c = 0)"
lookup_env_restr: fixes x :: "'a"   and S :: "'a set"   and m :: "'a \<Rightarrow> 'b" assumes "x \<in> S" shows "(m f|` S) x = m x"
Q_int: shows "integrable (M2 \<Otimes>\<^sub>M M1) (\<lambda>(x, y). f y x)"
LiberalConseq_sound: fixes P :: "'d set"   and \<Gamma> :: "'f \<Rightarrow> ('d, 'f, 'e) com option"   and \<Theta> :: "('d set \<times> 'f \<times> 'd set \<times> 'd set) set"   and F :: "'e set"   and c :: "('d, 'f, 'e) com"   and Q :: "'d set"   and A :: "'d set"   and n :: "nat" assumes "\<forall>s\<in>P. \<forall>t. \<exists>P' Q' A'. (\<forall>n. \<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P' c Q',A') \<and> ((s \<in> P' \<longrightarrow> t \<in> Normal ` Q' \<union> Abrupt ` A') \<longrightarrow> t \<in> Normal ` Q \<union> Abrupt ` A)" shows "\<Gamma>,\<Theta>\<Turnstile>n:\<^bsub>/F\<^esub> P c Q,A"
inv_join: fixes l :: "('b \<times> color) tree"   and r :: "('b \<times> color) tree"   and x :: "'b" assumes "invc l"   and "invh l"   and "invc r"   and "invh r" shows "invc (Set2_Join_RBT.join l x r) \<and> invh (Set2_Join_RBT.join l x r)"
trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s_fv_subst_subset: fixes t :: "('c, 'd) Term.term"   and F :: "(('c, 'd) Term.term \<times> ('c, 'd) Term.term) list"   and \<theta> :: "'d \<Rightarrow> ('c, 'e) Term.term" assumes "t \<in> trms\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s F" shows "fv (t \<cdot> \<theta>) \<subseteq> fv\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s (F \<cdot>\<^sub>p\<^sub>a\<^sub>i\<^sub>r\<^sub>s \<theta>)"
FAss1Throw1: fixes uf :: "bool"   and P :: "'addr J1_prog"   and t :: "'thread_id"   and a :: "'addr"   and F :: "String.literal"   and D :: "String.literal"   and e\<^sub>2 :: "'addr expr1"   and s :: "'heap \<times> 'addr val list" shows "uf,P,t \<turnstile>1 \<langle>Throw a\<bullet>F{D} := e\<^sub>2,s\<rangle> -\<lbrace>\<rbrace>\<rightarrow> \<langle>Throw a,s\<rangle>"
iMOD_minus: fixes x :: "nat"   and r :: "nat"   and m :: "nat"   and k :: "nat" assumes "x \<in> [ r, mod m ]"   and "k \<le> x - r" shows "(x - k \<in> [ r, mod m ]) = (k mod m = 0)"
"1_2": shows "exec_plan s as_1 = last slist_1"
l_set_tag_name_get_shadow_root_axioms_def: fixes set_tag_name_locs :: "(_) element_ptr \<Rightarrow> ((_) heap, exception, unit) prog set"   and get_shadow_root_locs :: "(_) element_ptr \<Rightarrow> ((_) heap \<Rightarrow> (_) heap \<Rightarrow> bool) set" shows "l_set_tag_name_get_shadow_root_axioms set_tag_name_locs get_shadow_root_locs \<equiv> \<forall>ptr h h' ptr'. \<forall>w\<in>set_tag_name_locs ptr. h \<turnstile> w \<rightarrow>\<^sub>h h' \<longrightarrow> (\<forall>r\<in>get_shadow_root_locs ptr'. r h h')"
assms_1: fixes x' :: "'a"   and y' :: "'a" assumes "x' \<in> set xs"   and "y' \<in> set xs"   and "rel x' y'" shows "x' = y'"
boundOutputChainEq': fixes xvec :: "name list"   and M :: "'d"   and P :: "('d, 'e, 'f) psi"   and yvec :: "name list"   and N :: "'d"   and Q :: "('d, 'e, 'f) psi" assumes "\<lparr>\<nu>*xvec\<rparr>M \<prec>' P = \<lparr>\<nu>*yvec\<rparr>N \<prec>' Q"   and "xvec \<sharp>* yvec" shows "\<exists>p. set p \<subseteq> set xvec \<times> set yvec \<and> distinctPerm p \<and> M = p \<bullet> N \<and> P = p \<bullet> Q \<and> xvec \<sharp>* N \<and> xvec \<sharp>* Q \<and> yvec \<sharp>* M \<and> yvec \<sharp>* P"
domain: shows "Domainp cr_int_poly_factorization_algorithm = (\<lambda>alg. \<forall>f fs. square_free f \<longrightarrow> 0 < degree f \<longrightarrow> alg f = fs \<longrightarrow> f = prod_list fs \<and> (\<forall>fi\<in>set fs. irreducible\<^sub>d fi))"
dvd: fixes a :: "nat"   and b :: "nat" shows "Suc a * (fact a * fact b) dvd fact (Suc (a + b))"
trg_src: fixes \<mu> :: "'a" assumes "arr \<mu>" shows "trg (src \<mu>) = src \<mu>"
class_add_is_type: fixes P :: "jvm_prog"   and T :: "ty"   and C :: "char list"   and cdec :: "jvm_method class" assumes "is_type P T" shows "is_type (class_add P (C, cdec)) T"
cong_modulus_mult: fixes divide :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and one :: "'a"   and modulo :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and euclidean_size :: "'a \<Rightarrow> nat"   and uminus :: "'a \<Rightarrow> 'a"   and division_segment :: "'a \<Rightarrow> 'a"   and x :: "'a"   and y :: "'a"   and m :: "'a"   and n :: "'a" assumes "class.unique_euclidean_ring divide plus minus zero times one modulo euclidean_size uminus division_segment"   and "unique_euclidean_semiring.cong modulo x y (times m n)" shows "unique_euclidean_semiring.cong modulo x y m"
acyclic_insert: fixes y :: "'b"   and x :: "'b"   and r :: "('b \<times> 'b) set" shows "acyclic (insert (y, x) r) = (acyclic r \<and> (x, y) \<notin> r\<^sup>*)"
min_dist_suc: fixes c :: "nat \<times> nat \<Rightarrow> 'a"   and v :: "nat"   and v' :: "nat"   and d :: "nat" assumes "Graph.connected c v v'"   and "Graph.min_dist c v v' = Suc d" shows "\<exists>v''. Graph.connected c v v'' \<and> (v'', v') \<in> Graph.E c \<and> Graph.min_dist c v v'' = d"
prop_in_thm: fixes v :: "i"   and F :: "\<Pi>\<^sub>1" shows "[Propositional F \<^bold>\<rightarrow> Indiscriminate F in v]"
path_Exit_source_1: fixes prog :: "cmd"   and as :: "(w_node \<times> (char list \<Rightarrow> val option) edge_kind \<times> w_node) list"   and n' :: "w_node" assumes "While_CFG.path prog (_Exit_) as n'" shows "n' = (_Exit_)"
build_split: fixes w :: "nat \<Rightarrow> 'b" shows "w = w 0 ## w [1..]"
concat_parts: fixes xss :: "'b list list"   and d :: "nat"   and i :: "nat" assumes "\<And>xs. xs \<in> set xss \<Longrightarrow> length xs = d"   and "i < length xss" shows "fixed_length_sublist (concat xss) d i = xss ! i"
cball_trivial: fixes x :: "'a" shows "cball x 0 = {x}"
surjective: fixes r :: "('a, 'b, 'c) configuration_scheme" shows "r = \<lparr>c_work = c_work r, c_pts = c_pts r, c_imp = c_imp r, \<dots> = more r\<rparr>"
map_tpOf_getTvars: fixes \<sigma>l :: "'tp list" shows "map tpOf (getTvars \<sigma>l) = \<sigma>l"
assms_5: assumes "a \<in># B"   and "b \<in># A" shows "P"
return_bind_spmf: fixes x :: "'c"   and f :: "'c \<Rightarrow> 'b spmf" shows "return_spmf x \<bind> f = f x"
tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p_is_comp_tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p: fixes a :: "('fun, ('fun, 'atom) Term.term \<times> nat) stateful_strand_step" shows "tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p a = comp_tfr\<^sub>s\<^sub>s\<^sub>t\<^sub>p \<Gamma> Pair a"
linear_poly_sum: fixes p :: "linear_poly"   and v :: "nat \<Rightarrow> 'a" shows "p \<lbrace> v \<rbrace> = (\<Sum>x\<in>vars p. coeff p x *R v x)"
assms_6: shows "(a + r + b) \<cdot> l \<le> l \<cdot> (a + r + b)"
of_nat_simps_1: shows "of_nat 0 = (0::'c)"
Weierstrass_trig_polynomial: fixes f :: "real \<Rightarrow> real"   and e :: "real" assumes "continuous_on {- pi..pi} f"   and "f (- pi) = f pi"   and "0 < e" obtains n :: "nat" and a :: "nat \<Rightarrow> real" and b :: "nat \<Rightarrow> real" where "\<And>x. x \<in> {- pi..pi} \<Longrightarrow> \<bar>f x - (\<Sum>k\<le>n. a k * sin (real k * x) + b k * cos (real k * x))\<bar> < e"
Peirce: fixes H :: "fmla set"   and A :: "fmla" assumes "H \<turnstile> neg A IMP A" shows "H \<turnstile> A"
distinct_mset_mset_set: fixes A :: "'c set" shows "distinct_mset (mset_set A)"
simps_1: fixes v :: "('a, 'b option) insert_message list" shows "is_certified_associated_string' (Inr v) = (set v = to_woot_character (M \<union> {Insert m}) ` insert_messages (M \<union> {Insert m}) \<and> sorted_wrt (<) (map a (ext_ids v)))"