hcju/isabelleps huggingface dataset:
hsnd_conv: fixes a :: "hf"   and b :: "hf" shows "hsnd \<langle>a, b\<rangle> = b"
case_2: fixes f1 :: "'c \<Rightarrow> 'b"   and f2 :: "'b" shows "(case bot.Bot of bot.Value x \<Rightarrow> f1 x | bot.Bot \<Rightarrow> f2) = f2"
False: shows "\<not> fst (fmrestrict_set vs (fst h), fmrestrict_set vs (snd h)) \<subseteq>\<^sub>f s"
IH: assumes "labels prog l c" shows "\<exists>l'. prog \<turnstile> \<langle>c,s,l\<rangle> \<leadsto>* \<langle>c'',s'',l'\<rangle> \<and> labels prog l' c''"
from_nat_le: fixes i :: "'a"   and k :: "nat" assumes "mod_type_class.to_nat i < k"   and "k < CARD('a)" shows "i < mod_type_class.from_nat k"
64239: fixes x :: "'a"   and y :: "'a" shows "- (x \<squnion> (- - x \<squnion> y)) = - (x \<squnion> y)"
rtrancl_len_E: fixes x :: "'b"   and y :: "'b"   and r :: "('b \<times> 'b) set" assumes "(x, y) \<in> r\<^sup>*" obtains n :: "nat" where "(x, y) \<in> r ^^ n"
that: assumes "normal s1 \<Longrightarrow> nrm A \<subseteq> dom (locals (snd s1))"   and "\<And>l. \<lbrakk>fst s1 = Some (Jump (Break l)); normal s0\<rbrakk> \<Longrightarrow> brk A l \<subseteq> dom (locals (snd s1))"   and "\<lbrakk>fst s1 = Some (Jump Ret); normal s0\<rbrakk> \<Longrightarrow> Result \<in> dom (locals (snd s1))" shows "thesis"
sub_block_upd_fresh: fixes i :: "'d"   and block :: "('e, 'd) fm list \<times> 'd"   and f :: "'d \<Rightarrow> 'f"   and j :: "'f" assumes "i \<notin> block_nominals block" shows "sub_block (f(i := j)) block = sub_block f block"
min_Suc_Suc: fixes m :: "nat"   and n :: "nat" shows "min (Suc m) (Suc n) = Suc (min m n)"
emeasure_dens_ctxt_measure_insert: fixes F :: "(nat \<Rightarrow> val) \<Rightarrow> val measure"   and t :: "pdf_type"   and f :: "(nat \<Rightarrow> val) \<Rightarrow> val \<Rightarrow> ennreal"   and \<rho> :: "nat \<Rightarrow> val"   and X :: "(nat \<Rightarrow> val) set" assumes "has_parametrized_subprob_density (state_measure (V \<union> V') \<Gamma>) F (stock_measure t) f"   and "\<rho> \<in> space (state_measure (Suc ` V') (case_nat t \<Gamma>))"   and "X \<in> sets (dens_ctxt_measure (shift_var_set V, Suc ` V', case_nat t \<Gamma>, insert_dens V V' f \<delta>) \<rho>)" shows "emeasure (dens_ctxt_measure (shift_var_set V, Suc ` V', case_nat t \<Gamma>, insert_dens V V' f \<delta>) \<rho>) X = \<integral>\<^sup>+ x. insert_dens V V' f \<delta> (merge (shift_var_set V) (Suc ` V') (x, \<rho>)) * indicator X (merge (shift_var_set V) (Suc ` V') (x, \<rho>)) \<partial>state_measure (shift_var_set V) (case_nat t \<Gamma>)"
bl_to_bin_rep_F: fixes n :: "nat"   and bl :: "bool list" shows "bl_to_bin (replicate n False @ bl) = bl_to_bin bl"
ce_rel_lm_9: fixes x :: "nat"   and y :: "nat"   and r :: "(nat \<times> nat) set" assumes "(x, y) \<in> r" shows "c_pair x y \<in> ce_rel_to_set r"
strictly_preserves_\<alpha>_ide: fixes a :: "'c"   and b :: "'c"   and c :: "'c" assumes "C.ide a"   and "C.ide b"   and "C.ide c" shows "F \<a>\<^sub>C[a, b, c] = \<a>\<^sub>D[F a, F b, F c]"
set_disconnected_nodes_typess_preserved: fixes w :: "((_) heap, exception, unit) prog"   and object_ptr :: "(_) document_ptr"   and h :: "(_) heap"   and h' :: "(_) heap" assumes "w \<in> set_disconnected_nodes_locs object_ptr"   and "h \<turnstile> w \<rightarrow>\<^sub>h h'" shows "DocumentClass.type_wf h = DocumentClass.type_wf h'"
FAIL_leof: fixes m :: "'c nres" shows "FAIL \<le>\<^sub>n m"
conversep_mono: fixes r :: "'c \<Rightarrow> 'b \<Rightarrow> bool"   and s :: "'c \<Rightarrow> 'b \<Rightarrow> bool" shows "(r\<inverse>\<inverse> \<le> s\<inverse>\<inverse>) = (r \<le> s)"
hm_it_has_next_def: fixes it :: "nat \<times> ('c \<times> 'd) list \<times> ('c, 'd) hashtable" shows "hm_it_has_next it \<equiv> return (case it of (0, [], x) \<Rightarrow> False | (0, aa # list, x) \<Rightarrow> True | (Suc nat, b) \<Rightarrow> True)"
lex_ord_raw_def: fixes ko :: "'o"   and f :: "'a \<Rightarrow> 'd \<Rightarrow> 'e \<Rightarrow> order option"   and xs :: "('a \<times> 'd) list \<times> 'o"   and ys :: "('a \<times> 'e) list \<times> 'o" shows "lex_ord_raw ko f xs ys = lex_ord_pair ko f (sort_oalist_aux ko xs) (sort_oalist_aux ko ys)"
simps_7: fixes f1 :: "'b"   and f2 :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and nat :: "nat" shows "rec_nat f1 f2 (Suc nat) = f2 nat (rec_nat f1 f2 nat)"
distinct_9: fixes x6 :: "ty" shows "Void \<noteq> x6\<lfloor>\<rceil>"
cong: fixes K :: "'a \<Rightarrow> 'a pmf set"   and S :: "'a set"   and S1 :: "'a set"   and S2 :: "'a set" shows "Reachability_Problem.p K S S1 S2 \<equiv> Reachability_Problem.p K S S1 S2"
add_new_mmuaux_eq: shows "add_new_mmuaux args rel1 rel2 nt aux = (tp + 1, tss', len + 1, maskL, maskR, a1_map', a2_map', done, done_length)"
insert_remove: fixes A :: "'c set"   and g :: "'c \<Rightarrow> 'd"   and x :: "'c" assumes "finite A" shows "sup_monoid.sum g (insert x A) = g x \<squnion> sup_monoid.sum g (A - {x})"
eq: shows "set (map (consistent_sign_vec_copr qs) (characterize_root_list_p p)) = set (map (consistent_sign_vec_copr qs) (characterize_root_list_p p)) \<inter> {s. z I s = 1} \<union> set (map (consistent_sign_vec_copr qs) (characterize_root_list_p p)) \<inter> {s. z I s = - 1}"
Sup_le_iff: fixes A :: "('c \<times> 'd) set"   and b :: "'c \<times> 'd" shows "less_eq_prod (Sup_prod A) b = (\<forall>a\<in>A. less_eq_prod a b)"
leg1_in_hom_1: shows "\<guillemotleft>g : src f \<rightarrow> trg r\<guillemotright>"
typ_of_eq_ax_6: shows "typ_of eq_combination_ax = Some propT"
is_extensional: fixes f :: "'a" assumes "\<not> arr f" shows "f \<otimes> \<I> = null"
IDE_implies_ide: fixes a :: "'c local.term set" assumes "IDE a" shows "ide a"
bexE: fixes A :: "'h set"   and P :: "'h \<Rightarrow> bool" assumes "\<exists>x\<in>A. P x" obtains x :: "'h" where "x \<in> A" and "P x"
is_regular_mult_sheaf_spec: fixes s :: "'a set \<Rightarrow> ('a \<times> 'a) set"   and U :: "'a set set"   and s' :: "'a set \<Rightarrow> ('a \<times> 'a) set" assumes "is_regular s U"   and "is_regular s' U"   and "U \<subseteq> Spec" shows "is_regular (mult_sheaf_spec U s s') U"
to_g: fixes x :: "'a" assumes "x \<in> cbox a b" shows "(\<lambda>n. f n x) \<longlonglongrightarrow> g x"
arr_eval_Arr: fixes t :: "'c local.term" assumes "Arr t" shows "arr \<lbrace>t\<rbrace>"
read_only_reads_unowned: fixes i :: "nat"   and j :: "nat"   and p\<^sub>i :: "'a"   and is\<^sub>i :: "instr list"   and \<theta>\<^sub>i :: "nat \<Rightarrow> nat option"   and sb\<^sub>i :: "'a memref list"   and \<D>\<^sub>i :: "bool"   and \<O>\<^sub>i :: "nat set"   and \<R>\<^sub>i :: "nat \<Rightarrow> bool option"   and p\<^sub>j :: "'a"   and is\<^sub>j :: "instr list"   and \<theta>\<^sub>j :: "nat \<Rightarrow> nat option"   and sb\<^sub>j :: "'a memref list"   and \<D>\<^sub>j :: "bool"   and \<O>\<^sub>j :: "nat set"   and \<R>\<^sub>j :: "nat \<Rightarrow> bool option" assumes "i < length ts"   and "j < length ts"   and "i \<noteq> j"   and "ts ! i = (p\<^sub>i, is\<^sub>i, \<theta>\<^sub>i, sb\<^sub>i, \<D>\<^sub>i, \<O>\<^sub>i, \<R>\<^sub>i)"   and "ts ! j = (p\<^sub>j, is\<^sub>j, \<theta>\<^sub>j, sb\<^sub>j, \<D>\<^sub>j, \<O>\<^sub>j, \<R>\<^sub>j)" shows "(\<O>\<^sub>j \<union> all_acquired sb\<^sub>j) \<inter> read_only_reads (acquired True (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) \<O>\<^sub>i) (dropWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb\<^sub>i) = {}"
assms: shows "A \<in> Units (ring_mat TYPE('a) n b)"
NEG_intro_2: fixes N :: "ntrm"   and B :: "ty" assumes "N \<in> NEGn B (\<parallel><B>\<parallel>)" shows "N \<in> \<parallel>(B)\<parallel>"
compare_GT_iff_LT: fixes x :: "'a"   and y :: "'a" shows "(compare\<cdot>x\<cdot>y = GT) = (compare\<cdot>y\<cdot>x = LT)"
apartment_trivial_morphism: fixes Xa :: "'b set set"   and \<A> :: "'b set set set"   and X :: "'b set set" assumes "ChamberComplexWithApartmentSystem Xa \<A>"   and "X \<in> \<A>" shows "ChamberComplexMorphism X X id"
assms_4: shows "\<forall>k op. k < length (trace_parallel_plan_strips (\<Pi>\<^sub>I) \<pi>) - 1 \<longrightarrow> \<A> (Operator k (index (\<Pi>\<^sub>\<O>) op)) = (op \<in> set (\<pi> ! k))"
split: fixes P :: "'a \<Rightarrow> bool"   and f1 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and f2 :: "nat \<Rightarrow> nat \<Rightarrow> 'a"   and msg :: "msg" shows "P (case msg of Pkt x xa \<Rightarrow> f1 x xa | Newpkt x xa \<Rightarrow> f2 x xa) = ((\<forall>x11 x12. msg = Pkt x11 x12 \<longrightarrow> P (f1 x11 x12)) \<and> (\<forall>x21 x22. msg = Newpkt x21 x22 \<longrightarrow> P (f2 x21 x22)))"
eff_enabled'_correct: fixes s :: "nat \<Rightarrow> nat option"   and eff :: "(nat \<times> nat) list \<times> nat \<times> nat option \<times> nat" assumes "s \<in> valid_states"   and "case eff of (pres, x, xa, xb) \<Rightarrow> wf_partial_state pres" shows "eff_enabled' s eff = eff_enabled s eff"
IMP_POST_NOCYC_AUX: fixes blues :: "'v set"   and reds :: "'v set"   and pr :: "'v list"   and pl :: "'v list" assumes "add_inv blues reds {}"   and "gen_dfs_outer E reachable V0 {} blues False"   and "is_lasso_prpl (pr, pl)" shows "False"
domain_eq: shows "Domainp pcr_rat = (\<lambda>x. ratrel x x)"
in_set_spmf: fixes x :: "'a"   and p :: "'a spmf" shows "(x \<in> set_spmf p) = (Some x \<in> set_pmf p)"
alwaysActive: fixes k :: "cnf" shows "\<exists>id. \<parallel>id\<parallel>\<^bsub>k\<^esub>"
axioms_1: fixes ext :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a list \<Rightarrow> 'a list \<Rightarrow> bool" assumes "ext_trans_before_irrefl ext" shows "ext_trans ext"
le1: shows "aff_dim (affine hull S) \<le> 1"
Q': shows "Q' \<in> carrier_mat (n - 1) (n - 1)"
div_add1_eq_if: fixes m :: "nat"   and a :: "nat"   and b :: "nat" assumes "0 < m" shows "(a + b) div m = a div m + b div m + (if a mod m + b mod m < m then 0 else Suc 0)"
map_empty_def: shows "Map_To_Mapping.map_empty = Map.empty"
right_minus: fixes a :: "'a" shows "a + - a = (0::'a)"
2: fixes ys :: "['a]"   and a :: "'a" shows "\<forall>i\<ge>0. ys !! MkI\<cdot>i = (nth\<cdot>(a : ys) oo (+1))\<cdot>(MkI\<cdot>i)"
set_eq: shows "{p. p partitions Suc m \<and> sum p {..Suc m} = Suc k} = {p. p partitions Suc m \<and> sum p {..Suc m} = Suc k \<and> p 1 \<noteq> 0} \<union> {p. p partitions Suc m \<and> sum p {..Suc m} = Suc k \<and> p 1 = 0}"
distrib_left_NO_MATCH: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and x :: "'c"   and y :: "'c"   and a :: "'b"   and b :: "'b"   and c :: "'b" assumes "class.semiring plus times"   and "NO_MATCH (x div y) a" shows "times a (plus b c) = plus (times a b) (times a c)"
ComplI: fixes c :: "'d"   and A :: "'d set" assumes "c \<in> A \<Longrightarrow> False" shows "c \<in> - A"
span_unit_vecs_is_carrier: shows "local.span (set (unit_vecs n)) = carrier_vec n"
real_scalar_mult2: fixes x :: "'a"   and A :: "'a" shows "real_of (inner_field x x) *\<^sub>R A = inner_field x x * A"
degree_pCons_le: fixes a :: "'b"   and p :: "'b poly" shows "degree (pCons a p) \<le> Suc (degree p)"
axioms_1: fixes \<alpha> :: "V"   and \<AA> :: "V"   and \<BB> :: "V"   and \<FF> :: "V" assumes "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^sub>.\<^sub>i\<^sub>s\<^sub>o\<^bsub>\<alpha>\<^esub> \<BB>" shows "\<FF> : \<AA> \<mapsto>\<mapsto>\<^sub>D\<^sub>G\<^bsub>\<alpha>\<^esub> \<BB>"
diff_add_assoc2: fixes a :: "'k multiset"   and b :: "'k multiset"   and c :: "'k multiset" assumes "a \<subseteq># b" shows "b + c - a = b - a + c"
P'_def: shows "P' = (\<exists>b. (with (0::'a) zero' plus' scale' : \<guillemotleft>independent\<guillemotright> b) \<and> (with zero' plus' scale' : \<guillemotleft>span\<guillemotright> b) = (with zero' plus' scale' : \<guillemotleft>span\<guillemotright> V'))"
wp_Un_subset: fixes act :: "('b \<times> 'b) set"   and A :: "'b set"   and B :: "'b set" shows "wp act A \<union> wp act B \<subseteq> wp act (A \<union> B)"
less_le: fixes x :: "nat list"   and y :: "nat list" shows "(x <\<^sub>v y) = (x \<le>\<^sub>v y \<and> x \<noteq> y)"
divisors_pos_funD: fixes df :: "'a \<Rightarrow> 'a list"   and x :: "'a"   and d :: "'a" assumes "divisors_pos_fun df"   and "x \<noteq> (0::'a)"   and "d dvd x"   and "(0::'a) < d" shows "d \<in> set (df x)"
A11: shows "\<forall>X Y. (\<exists>a. \<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B a x y) \<longrightarrow> (\<exists>b. \<forall>x y. x \<in> X \<and> y \<in> Y \<longrightarrow> B x b y)"
length_last_fusion_def: fixes g :: "('d, 'e) generator"   and s :: "'e" shows "length_last_fusion g s = length_last (list.unfoldr g s)"
dim_subset: fixes S :: "'a set"   and T :: "'a set" assumes "S \<subseteq> T" shows "dim S \<le> dim T"
simps_5: fixes v :: "('a, 'b, 'c, 'd) Tree"   and va :: "('a, 'b, 'c, 'd) Tree set['a]" shows "(tNot v =\<^sub>\<alpha> tConj va) = False"
inorder_insert: fixes \<Delta>1 :: "nat"   and \<Delta>2 :: "nat"   and \<Gamma>1 :: "nat"   and \<Gamma>2 :: "nat"   and t :: "('b \<times> nat) tree"   and x :: "'b" assumes "WBT \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2"   and "Sorted_Less.sorted (Tree2.inorder t)" shows "Tree2.inorder (WBT.insert \<Delta>1 \<Delta>2 \<Gamma>1 \<Gamma>2 x t) = ins_list x (Tree2.inorder t)"
continuous_log: fixes S :: "complex set"   and f :: "complex \<Rightarrow> complex" assumes "SC_Chain S"   and "simply_connected S"   and "continuous_on S f"   and "\<And>z. z \<in> S \<Longrightarrow> f z \<noteq> 0" shows "\<exists>g. continuous_on S g \<and> (\<forall>z\<in>S. f z = exp (g z))"
simps_1: shows "SkewBinomialHeapStruc.queue_to_multiset [] = {#}"
effect_guardE: fixes P :: "heap \<Rightarrow> bool"   and f :: "heap \<Rightarrow> 'a \<times> heap"   and h :: "heap"   and h' :: "heap"   and r :: "'a" assumes "effect (Heap_Monad.guard P f) h h' r" obtains "h' = snd (f h)" and "r = fst (f h)" and "P h"
fixp_def: fixes Sup :: "'b set \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and f :: "'b \<Rightarrow> 'b" assumes "class.ccpo Sup less_eq less" shows "ccpo.fixp Sup less_eq f = Sup (ccpo.iterates Sup less_eq f)"
moebius_mu_0: shows "moebius_mu 0 = (0::'b)"
simps_5: fixes f1 :: "char list option \<Rightarrow> 'd \<Rightarrow> 'c"   and f2 :: "char list option \<Rightarrow> 'e \<Rightarrow> 'c"   and x11 :: "char list option"   and x12 :: "'d" shows "(case sum.Inl x11 x12 of sum.Inl x xa \<Rightarrow> f1 x xa | sum.Inr x xa \<Rightarrow> f2 x xa) = f1 x11 x12"
fup_Aexp_edom: fixes e :: "exp"   and a :: "Arity\<^sub>\<bottom>" shows "edom (\<A>\<^sup>\<bottom>\<^bsub>a\<^esub> e) \<subseteq> fv e"
check_poly_strict_mono_smart_def: fixes discrete :: "bool"   and pm :: "bool"   and gt :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and p :: "('d monom \<times> 'c) list"   and v :: "'d" shows "check_poly_strict_mono_smart discrete pm gt p v \<equiv> if discrete then check_poly_strict_mono_discrete gt p v else check_poly_strict_mono pm p v"
intro: fixes S :: "'a set"   and X :: "'a set" assumes "S \<subseteq> X" shows "ind_topology_axioms X S"
avl_insert_1: fixes t :: "('b \<times> nat) tree"   and x :: "'b" assumes "avl t" shows "avl (AVL_Set_Code.insert x t)"
hd: shows "{x1 EQ x1'} \<turnstile> P x1 x2 x3 x4 IFF P x1' x2 x3 x4"
\<psi>_char_1: shows "\<guillemotleft>F'.\<psi> : D'.unity \<rightarrow>\<^sub>D F C'.unity\<guillemotright>"
twoCisTypeI_7: shows "(\<lambda>x. twoC (x, 0)) = (\<lambda>x. (a + (b - a) * x, g2 (a + (b - a) * x)))"
refl_on_def: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" shows "refl_on A r = (r \<subseteq> A \<times> A \<and> (\<forall>x\<in>A. x \<succeq>[r] x))"
validS_def: fixes fs :: "formula list" shows "validS fs = (\<forall>M. \<forall>phi\<in>modelAssigns M. evalS M phi fs = True)"
is_num_normalize_1: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and uminus :: "'a \<Rightarrow> 'a"   and a :: "'a"   and b :: "'a"   and c :: "'a" assumes "class.neg_numeral minus plus zero uminus" shows "plus (plus a b) c = plus a (plus b c)"
integrable_on_powr_from_0': fixes a :: "real"   and c :: "real" assumes "- 1 < a"   and "0 \<le> c" shows "(\<lambda>x. x powr a) integrable_on {0<..c}"
A_5: shows "Z \<subseteq> {u \<in> V. u x - u y \<le> real_of_int m}"
subst_bvs_extend_lower_level: fixes a :: "Core.term"   and args :: "Core.term list"   and t :: "Core.term" assumes "\<forall>x\<in>set (a # args). is_closed x" shows "subst_bv a (subst_bvs1' t 1 args) = subst_bvs (a # args) t"
r_minus: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c"   and y :: "'c" assumes "ring R"   and "x \<in> carrier R"   and "y \<in> carrier R" shows "x \<otimes>\<^bsub>R\<^esub> \<ominus>\<^bsub>R\<^esub> y = \<ominus>\<^bsub>R\<^esub> (x \<otimes>\<^bsub>R\<^esub> y)"
match_wf_tuple: fixes f :: "nat \<Rightarrow> 'a option"   and ts :: "'a MFOTL.trm list"   and xs :: "'a list"   and n :: "nat" assumes "Some f = match ts xs" shows "wf_tuple n (\<Union> (fv_trm ` set ts)) (tabulate f 0 n)"
add_ac_1: fixes a :: "'c"   and b :: "'c"   and c :: "'c" shows "a + b + c = a + (b + c)"
admissible_targets': fixes l :: "'s"   and g :: "('c, real) cconstraint"   and \<mu> :: "('c set \<times> 's) pmf"   and X :: "'c set"   and l' :: "'s"   and R :: "('c \<Rightarrow> real) set" assumes "(l, g, \<mu>) \<in> PTA.trans_of A"   and "(X, l') \<in> set_pmf \<mu>"   and "R \<subseteq> {v. v \<turnstile> g}" shows "[X::=0]R \<subseteq> {v. v \<turnstile> PTA.inv_of A l'}"
Re_eq_pihalf_lemma: fixes z :: "complex" assumes "\<bar>Re z\<bar> = pi / 2"   and "Im z = 0" shows "Re ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2) = 0 \<and> 0 \<le> Im ((exp (\<i> * z) + inverse (exp (\<i> * z))) / 2)"
inv_def: shows "inv \<equiv> \<lambda>n. inverse (real n)"
eq_expexpgen: fixes X :: "cmsg"   and Y :: "cmsg" assumes "eq X Y" shows "\<forall>X' X''. X = cExp (cExp cGen X') X'' \<longrightarrow> (\<exists>Y' Y''. Y = cExp (cExp cGen Y') Y'' \<and> (eq X' Y' \<and> eq X'' Y'' \<or> eq X' Y'' \<and> eq X'' Y'))"
has_bochner_integral_inner_left: fixes c :: "'c"   and M :: "'d measure"   and f :: "'d \<Rightarrow> 'c"   and x :: "'c" assumes "c \<noteq> (0::'c) \<Longrightarrow> has_bochner_integral M f x" shows "has_bochner_integral M (\<lambda>x. f x \<bullet> c) (x \<bullet> c)"
conga_diff2: fixes Bet :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and Cong :: "'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool"   and A :: "'a"   and B :: "'a"   and C :: "'a"   and A' :: "'a"   and B' :: "'a"   and C' :: "'a" assumes "Tarski_neutral_dimensionless Bet Cong"   and "Tarski_neutral_dimensionless.CongA Bet Cong A B C A' B' C'" shows "C \<noteq> B"
f_def: shows "f \<equiv> let k = Cn 2 r_pdec1 [recf.Id 2 0]; r = Cn 2 r_result1 [Cn 2 r_pdec2 [recf.Id 2 0], Cn 2 r_s [Cn 2 r_append_zeros [recf.Id 2 1, k]], Cn 2 r_add [Cn 2 r_length [recf.Id 2 1], k]] in Cn 2 r_eq [r, r_constn 1 1]"
nat_ge2_conv: fixes n :: "nat" shows "(2 \<le> n) = (n \<noteq> 0 \<and> n \<noteq> 1)"
coeff_const: fixes a :: "'b"   and i :: "nat" shows "coeff [:a:] i = (if i = 0 then a else (0::'b))"
enc_and_TRel_impl_indRelRTPO_respects_success: fixes success :: "'barbs"   and TRel :: "('procT \<times> 'procT) set" assumes "(\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x))) \<and> (\<forall>S x. x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S TargetTerm (\<lbrakk>S\<rbrakk>) \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T TargetTerm (\<lbrakk>S\<rbrakk>) \<and> T\<down><TWB>x)) \<longrightarrow> x \<in> {success} \<and> ((\<exists>Sa. Sa \<in>S SourceTerm S \<and> Sa\<down><SWB>x) \<or> (\<exists>T. T \<in>T SourceTerm S \<and> T\<down><TWB>x)))"   and "rel_preserves_barb_set TRel TWB {success}"   and "rel_reflects_barb_set TRel TWB {success}" shows "rel_respects_barb_set (indRelRTPO TRel) (STCalWB SWB TWB) {success}"
comp_in_homI: fixes f :: "'a"   and a :: "'a"   and b :: "'a"   and g :: "'a"   and c :: "'a" assumes "\<guillemotleft>f : a \<rightarrow> b\<guillemotright>"   and "\<guillemotleft>g : b \<rightarrow> c\<guillemotright>" shows "\<guillemotleft>C g f : a \<rightarrow> c\<guillemotright>"
countable_support: shows "countable {x. M.prob {x} \<noteq> 0}"
le_finE: fixes w\<^sub>1 :: "'item list"   and w\<^sub>2 :: "'item list" assumes "w\<^sub>1 \<preceq>\<^sub>F w\<^sub>2" obtains v\<^sub>1 :: "'item list" where "w\<^sub>1 @ v\<^sub>1 =\<^sub>F w\<^sub>2"
simps_3: fixes t :: "('fun, 'var) Term.term"   and S :: "('fun, 'var) extstrand_step list" shows "to_st (Decomp t # S) = decomp t @ to_st S"
singleton_list_lengthE: fixes xs :: "'b list" assumes "length xs = 1" obtains x :: "'b" where "xs = [x]"
connect': assumes "x \<in> closure S"   and "x \<in> closure T" shows "f' x = g' x"
exhaust: fixes y :: "Value" obtains x1 :: "bool" where "y = boolV x1"   | x2 :: "int" where "y = intgV x2"   | x3 :: "int" where "y = shortV x3"   | x4 :: "int" where "y = byteV x4"   | x51 :: "CTypeId" and x52 :: "nat" where "y = objV x51 x52"   | x61 :: "Arraytype" and x62 :: "nat" where "y = arrV x61 x62"   | "y = nullV"
conv: shows "converges_to (\<lambda>i. bnd_get ((bnd_update ^^ i) init)) x"
functions_of_is_surj_on: fixes A :: "'c set"   and M :: "'d multiset"   and B :: "'d set" assumes "finite A"   and "size M = card A"   and "set_mset M = B" shows "univ (\<lambda>f. f ` A = B) (functions_of A M)"
lt_tail: fixes p :: "'a \<Rightarrow>\<^sub>0 'c" assumes "punit.tail p \<noteq> 0" shows "punit.lt (punit.tail p) \<prec> punit.lt p"
pseudo_mod_left_0: fixes x :: "'b poly" shows "pseudo_mod 0 x = 0"
prev_shadow_root: shows "h \<turnstile> get_shadow_root element_ptr \<rightarrow>\<^sub>r None"
HCollect_iff: fixes x :: "hf"   and P :: "hf \<Rightarrow> bool"   and A :: "hf" shows "(x \<^bold>\<in> HCollect P A) = (P x \<and> x \<^bold>\<in> A)"
WTrts_hext_mono: fixes P :: "(char list list \<times> char list exp) prog"   and h :: "heap"   and sh :: "sheap"   and E :: "char list \<Rightarrow> ty option"   and es :: "char list exp list"   and Ts :: "ty list"   and h' :: "heap" assumes "P,E,h,sh \<turnstile> es [:] Ts"   and "h \<unlhd> h'" shows "P,E,h',sh \<turnstile> es [:] Ts"
smooth_on_norm: fixes k :: "enat"   and S :: "'b set"   and f :: "'b \<Rightarrow> 'c" assumes "k-smooth_on S f"   and "(0::'c) \<notin> f ` S"   and "open S" shows "k-smooth_on S (\<lambda>x. norm (f x))"
r_tuple_encode: fixes xs :: "nat list"   and n :: "nat" assumes "length xs = Suc n" shows "eval (r_tuple_encode n) xs \<down>= tuple_encode n xs"
in_set_conv_decomp_last: fixes x :: "'d"   and xs :: "'d list" shows "(x \<in> set xs) = (\<exists>ys zs. xs = ys @ x # zs \<and> x \<notin> set zs)"
zero_least_kA: shows "is_zero_row_upt_k (LEAST n. A $ n $ mod_type_class.from_nat k \<noteq> (0::'a) \<and> mod_type_class.from_nat i \<le> n) k A"
not_le: fixes x :: "'t"   and y :: "'t" shows "(\<not> le_of_nat_term_order cmp_term x y) = lt_of_nat_term_order cmp_term y x"
tensor_preserves_ide: fixes a :: "'d"   and b :: "'d" assumes "D.ide a"   and "D.ide b" shows "D.ide (a \<otimes>\<^sub>D b)"
eq2: shows "W Q (Suc k) = adjoint M0 * Q * M0 + adjoint M1 * WS (W Q k) * M1"
PrefixListHasTail: fixes l1 :: "'b list"   and l2 :: "'b list" assumes "prefixList l1 l2" shows "\<exists>l. l2 = l1 @ l \<and> l \<noteq> []"
pre_arith_simps_1: fixes b :: "'b"   and c :: "'b"   and a :: "'b" shows "(max b c \<le> a) = (b \<le> a \<and> c \<le> a)"
of_real_minus: fixes x :: "real" shows "of_real (- x) = - of_real x"
healthy_inf_cont: fixes A :: "'a hrel set" assumes "A \<subseteq> \<lbrakk>\<H>\<rbrakk>\<^sub>H"   and "\<not> A = {}" shows "\<^bold>\<Sqinter>A = \<Or>A"
get_child_nodes_ptr_in_heap: fixes h :: "(_) heap"   and ptr :: "(_) object_ptr" assumes "h \<turnstile> ok get_child_nodes ptr" shows "ptr |\<in>| object_ptr_kinds h"