hcju/isabelleps huggingface dataset:
fsyms_int_cong: fixes intT :: "'tp \<Rightarrow> 'b \<Rightarrow> bool"   and intF1 :: "'fsym \<Rightarrow> 'b list \<Rightarrow> 'b"   and intP :: "'psym \<Rightarrow> 'b list \<Rightarrow> bool"   and intF2 :: "'fsym \<Rightarrow> 'b list \<Rightarrow> 'b"   and T :: "'fsym trm"   and \<xi> :: "Preliminaries.var \<Rightarrow> 'b" assumes "Struct wtFsym wtPsym arOf resOf intT intF1 intP"   and "Struct wtFsym wtPsym arOf resOf intT intF2 intP"   and "\<And>f. f \<in> fsyms T \<Longrightarrow> intF1 f = intF2 f" shows "Struct.int intF1 \<xi> T = Struct.int intF2 \<xi> T"
label_wrtD': fixes B :: "'c set"   and f :: "'a \<Rightarrow> 'c"   and C :: "'a set" assumes "label_wrt B f"   and "chamber C" shows "bij_betw f C B"
simps: fixes n :: "nat" shows "nextX (X n) = X (Suc n)"
comp_assoc_tuple_1: fixes f0 :: "'a"   and a :: "'a"   and b0 :: "'a"   and f1 :: "'a"   and b1 :: "'a"   and f2 :: "'a"   and b2 :: "'a" assumes "\<guillemotleft>f0 : a \<rightarrow> b0\<guillemotright>"   and "\<guillemotleft>f1 : a \<rightarrow> b1\<guillemotright>"   and "\<guillemotleft>f2 : a \<rightarrow> b2\<guillemotright>" shows "\<a>[b0, b1, b2] \<cdot> \<langle>\<langle>f0, f1\<rangle>, f2\<rangle> = \<langle>f0, \<langle>f1, f2\<rangle>\<rangle>"
confluence_equiv_ChurchRosser: shows "is_confluent = is_ChurchRosser"
heap_is_wellformed_one_disc_parent: fixes h :: "(_) heap"   and document_ptr :: "(_) document_ptr"   and disc_nodes :: "(_) node_ptr list"   and document_ptr' :: "(_) document_ptr"   and disc_nodes' :: "(_) node_ptr list" assumes "heap_is_wellformed h"   and "h \<turnstile> get_disconnected_nodes document_ptr \<rightarrow>\<^sub>r disc_nodes"   and "h \<turnstile> get_disconnected_nodes document_ptr' \<rightarrow>\<^sub>r disc_nodes'"   and "set disc_nodes \<inter> set disc_nodes' \<noteq> {}" shows "document_ptr = document_ptr'"
assms_2: shows "S \<equiv> \<lambda>n. \<Sum>k | k \<in> {1..n} \<and> coprime k n. f k n"
cmp_simps_1: fixes f :: "'a"   and g :: "'a" assumes "C.ide f"   and "C.ide g"   and "src\<^sub>C f = trg\<^sub>C g" shows "D.arr (\<Phi> (f, g))"
Fract_less_one_iff: fixes b :: "int"   and a :: "int" assumes "0 < b" shows "(Fract a b < 1) = (a < b)"
splitDigit_inv': fixes p :: "'c \<Rightarrow> bool"   and i :: "'c"   and d :: "('b, 'c) Digit"   and l :: "('b, 'c) Node list"   and nd :: "('b, 'c) Node"   and r :: "('b, 'c) Node list"   and n :: "nat" assumes "splitDigit p i d = (l, nd, r)"   and "is_measured_digit d"   and "is_leveln_digit n d" shows "(\<forall>x\<in>set l. is_measured_node x \<and> is_leveln_node n x) \<and> (\<forall>x\<in>set r. is_measured_node x \<and> is_leveln_node n x) \<and> is_measured_node nd \<and> is_leveln_node n nd"
take_0: fixes xs :: "'b SList.list" shows "SList.take xs 0 = []"
Lp_Lq_mult_2: fixes p :: "ennreal"   and q :: "ennreal"   and r :: "ennreal"   and f :: "'b \<Rightarrow> real"   and M :: "'b measure"   and g :: "'b \<Rightarrow> real" assumes "1 / p + 1 / q = 1 / r"   and "f \<in> space\<^sub>N (\<LL> p M)"   and "g \<in> space\<^sub>N (\<LL> q M)" shows "Norm (\<LL> r M) (\<lambda>x. f x * g x) \<le> Norm (\<LL> p M) f * Norm (\<LL> q M) g"
pair_map_spmf1: fixes f :: "'g \<Rightarrow> 'e"   and p :: "'g spmf"   and q :: "'f spmf" shows "pair_spmf (map_spmf f p) q = map_spmf (apfst f) (pair_spmf p q)"
assoc_simps_2: fixes f :: "'a"   and g :: "'a"   and h :: "'a" assumes "S.ide f"   and "S.ide g"   and "S.ide h"   and "S.src f = S.trg g"   and "S.src g = S.trg h" shows "S.src (S.\<a> f g h) = S.src h"
memory_read_privilege: fixes s' :: "'b sparc_state"   and asi :: "word8"   and addr :: "word32"   and s :: "'b sparc_state" assumes "s' = snd (memory_read asi addr s) \<and> get_S (cpu_reg_val PSR s) = 0" shows "get_S (cpu_reg_val PSR s') = 0"
terminated_GST_stable: fixes q :: "'g set"   and sh' :: "('h, 'i, 'g, 'j, 'k) system_state"   and sh :: "('h, 'i, 'g, 'j, 'k) system_state"   and p :: "'g" assumes "system_step q sh' sh"   and "atLs p {{}} sh" shows "GST sh p = GST sh' p"
commutes_with_composition: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> A" shows "(g \<circ> f \<down> A) (multA x y) = multC ((g \<circ> f \<down> A) x) ((g \<circ> f \<down> A) y)"
DBM_zone_repr_def: fixes M :: "nat \<Rightarrow> nat \<Rightarrow> 'd DBMEntry"   and v :: "'b \<Rightarrow> nat"   and n :: "nat" shows "[M]\<^bsub>v,n\<^esub> = {u. u \<turnstile>\<^bsub>v,n\<^esub> M}"
simps: fixes a :: "'b"   and S :: "'b set" shows "(a \<in> \<langle>S\<rangle>) = (a = (0::'b) \<or> (\<exists>s. a = s \<and> s \<in> S) \<or> (\<exists>w w'. a = w - w' \<and> w \<in> \<langle>S\<rangle> \<and> w' \<in> \<langle>S\<rangle>))"
conv_abscissa_perzeta: fixes q :: "real" shows "conv_abscissa (fds_perzeta q) \<le> 1"
same_sgn_abs_add: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and sgn :: "'b \<Rightarrow> 'b"   and abs :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and b :: "'b"   and a :: "'b" assumes "class.linordered_idom plus zero minus uminus less_eq less sgn abs times one"   and "sgn b = sgn a" shows "abs (plus a b) = plus (abs a) (abs b)"
keys_homogenize: fixes x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "x \<notin> indets p" shows "keys (homogenize x p) = (\<lambda>t. monomial (poly_deg p - deg_pm t) x + t) ` keys p"
has_integral_periodic_offset_pos: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> real"   and i :: "real"   and c :: "real" assumes "has_bochner_integral (lebesgue_on {a..b}) f i"   and "\<And>x. f (x + (b - a)) = f x"   and "0 \<le> c"   and "a + c \<le> b" shows "has_bochner_integral (lebesgue_on {a..b}) (\<lambda>x. f (x + c)) i"
assms_2: fixes xs :: "'a option list" assumes "P xs" shows "P (None # xs)"
bangPres: fixes P :: "pi"   and Q :: "pi" assumes "P \<simeq> Q" shows "!P \<simeq> !Q"
is_prefix_bd_def: fixes xs :: "'b list"   and ys :: "'b list" shows "is_prefix_bd xs ys \<equiv> (if length xs \<le> length ys then WHILE\<^sub>T\<^bsup>case_sum (\<lambda>x. True) (\<lambda>s. s \<le> length xs \<and> take s xs = take s ys)\<^esup> (case_sum (\<lambda>x. False) (\<lambda>s. s < length xs)) (\<lambda>s. ASSERT (\<not> isl s) \<bind> (\<lambda>_. let s = projr s in ASSERT (s < length xs \<and> s < length ys) \<bind> (\<lambda>_. if xs ! s = ys ! s then RETURN (Inr (s + 1)) else RETURN (Inl False)))) (Inr 0) \<bind> case_sum (\<lambda>e. RETURN (Inl e)) (\<lambda>x. RETURN (Inl True)) else RETURN (Inl False)) \<bind> (\<lambda>r. case r of Inl e \<Rightarrow> RETURN (Inr e) | Inr r \<Rightarrow> RETURN (Inr r))"
fundamental_theorem_of_linear_inequalities_A_imp_D: fixes A :: "'b vec set"   and n :: "nat"   and b :: "'b vec" assumes "A \<subseteq> carrier_vec n"   and "finite A"   and "b \<in> gram_schmidt.cone n A" shows "\<nexists>c. c \<in> carrier_vec n \<and> (\<forall>a\<^sub>i\<in>A. (0::'b) \<le> c \<bullet> a\<^sub>i) \<and> c \<bullet> b < (0::'b)"
continuous_map_square_root: shows "continuous_map euclideanreal euclideanreal sqrt"
deg_pm_homogenize: fixes t :: "'b \<Rightarrow>\<^sub>0 nat"   and x :: "'b"   and p :: "('b \<Rightarrow>\<^sub>0 nat) \<Rightarrow>\<^sub>0 'c" assumes "t \<in> keys (homogenize x p)" shows "deg_pm t = poly_deg p"
def_lfp_induct_set: fixes A :: "'a set"   and f :: "'a set \<Rightarrow> 'a set"   and a :: "'a"   and P :: "'a \<Rightarrow> bool" assumes "A \<equiv> lfp f"   and "mono f"   and "a \<in> A"   and "\<And>x. x \<in> f (A \<inter> {x. P x}) \<Longrightarrow> P x" shows "P a"
conv_abscissa_cong: fixes f :: "'b fds"   and g :: "'b fds" assumes "\<forall>\<^sub>F n in sequentially. fds_nth f n = fds_nth g n" shows "conv_abscissa f = conv_abscissa g"
f: fixes x :: "'c" shows "spmf.mono_body (\<lambda>f. F f x)"
false_sup_2: fixes P :: "'b upred" shows "P \<sqinter> false = P"
conseq_3: fixes s :: "char list \<Rightarrow> int" shows "Q' s \<le> Q s"
isSimplePath_fwd: fixes s :: "nat"   and p :: "(nat \<times> nat) list"   and t :: "nat" shows "isSimplePath s p t = (isPath s p t \<and> distinct (pathVertices_fwd s p))"
incoming_def: fixes v :: "nat" shows "incoming v \<equiv> {(u, v) |u. (u, v) \<in> E}"
M'_def: shows "M' = max (nat \<lceil>a\<rceil>) M"
lookup_def: shows "lookup \<equiv> map_fun impl_of id rbt_lookup"
canonical_len: fixes n :: "nat"   and M :: "nat \<Rightarrow> nat \<Rightarrow> 'c"   and i :: "nat"   and j :: "nat"   and xs :: "nat list" assumes "canonical M n"   and "i \<le> n"   and "j \<le> n"   and "set xs \<subseteq> {0..n}" shows "M i j \<le> len M i j xs"
is_wt_instance_of_condD: fixes t :: "('fun, 'var) Term.term"   and s :: "('fun, 'var) Term.term" assumes "is_wt_instance_of_cond \<Gamma> t s" obtains \<delta> :: "'var \<Rightarrow> ('fun, 'var) Term.term" where "\<Gamma> t = \<Gamma> s" and "mgu t s = Some \<delta>" and "inj_on \<delta> (fv t)" and "\<delta> ` fv t \<subseteq> range Var"
assms: fixes \<phi> :: "'a ltln"   and \<psi> :: "'a ltln" shows "atoms_ltln (f \<phi> \<psi>) \<subseteq> atoms_ltln \<phi> \<union> atoms_ltln \<psi>"
SingletonInRangeExplode: fixes x :: "ZF" shows "{x} \<in> range explode"
SN_onI: fixes A :: "'b set"   and r :: "('b \<times> 'b) set" assumes "\<And>f. \<lbrakk>f 0 \<in> A; \<forall>i. (f i, f (Suc i)) \<in> r\<rbrakk> \<Longrightarrow> False" shows "SN_on r A"
inference_saturated_def: fixes S :: "'a eclause set" shows "inference_saturated S = (\<forall>C P \<sigma> C' D \<theta> \<eta>. derivable C P S \<sigma> Ground C' \<longrightarrow> ground_clause (cl_ecl C) \<longrightarrow> grounding_set P \<sigma> \<longrightarrow> derivable D P S \<theta> FirstOrder C' \<longrightarrow> trms_subsumes D C \<eta> \<longrightarrow> \<sigma> \<doteq> \<theta> \<lozenge> \<eta> \<longrightarrow> redundant_inference (subst_ecl D \<eta>) S P \<sigma>)"
dim_row_of_tensor_gate: 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 "dim_row (G1 \<Otimes> G2) = 2 ^ (m + n)"
wlsFSbSw_recAll_termFSwSbMorph: fixes MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'c, 'd) Recursion.model" assumes "wlsFSbSw MOD" shows "termFSwSbMorph (rec MOD) (recAbs MOD) MOD"
bin_rl_eqI: fixes x :: "int"   and y :: "int" assumes "x div 2 = y div 2"   and "odd x = odd y" shows "x = y"
ensures_partial_saturation: fixes S :: "'at Literal set set"   and E2 :: "'at"   and R :: "'at Literal set set" assumes "partial_saturation S E2 (S \<union> R)"   and "all_fulfill (\<lambda>x. \<not> tautology x) S"   and "all_fulfill (in_all_resolvents_upon S E2) R"   and "all_fulfill (\<lambda>x. \<not> redundant x S) R" shows "partial_saturation (S \<union> R) E2 (S \<union> R)"
assms_2: shows "(XS, X) \<in> \<langle>A\<rangle>list_wset_rel"
preserves_cod: fixes f :: "'a" assumes "C.arr f" shows "C.cod (trg\<^sub>C f) = trg\<^sub>C (C.cod f)"
bounded_iff: fixes b :: "'d"   and c :: "'d"   and a :: "'d" shows "(max b c \<le> a) = (b \<le> a \<and> c \<le> a)"
wf_digraph_axioms: shows "wf_digraph (app_iso hom G)"
extend_state_oracle2_def: fixes callee :: "'a \<Rightarrow> 'c \<Rightarrow> ('d \<times> 'a) spmf" shows "callee\<dagger> = (\<lambda>(s, s') x. map_spmf (\<lambda>(y, s). (y, s, s')) (callee s x))"
SystemDISubst_def: shows "SystemDISubst = \<lparr>SFunctions = \<lambda>f. if f = fid1 then Some ($f (Inr vid1) local.empty) else if f = fid2 then Some (Const 0) else None, SPredicates = \<lambda>p. if p = vid1 then Some TT else None, SContexts = Map.empty, SPrograms = Map.empty, SODEs = \<lambda>c. if c = vid1 then Some (OProd (OSing vid1 (f0 fid1)) (OSing vid2 (Var vid1))) else None\<rparr>"
assms: fixes j :: "'a \<Rightarrow> 'a" assumes "j permutes UNIV"   and "j \<noteq> id" shows "degree (\<Prod>i\<in>UNIV. to_fun A i (j i)) < degree (\<Prod>i\<in>UNIV. to_fun A i i)"
simps_1: shows "braun_indices \<langle>\<rangle> = {}"
S: shows "S \<in> carrier_mat (m + n) (m + n)"
bodyf: shows "\<forall>l\<in>dom f. body (the (f l))"
mono_step': fixes S1 :: "'a st option"   and S2 :: "'a st option"   and C1 :: "'a st option acom"   and C2 :: "'a st option acom" assumes "S1 \<le> S2"   and "C1 \<le> C2" shows "step' S1 C1 \<le> step' S2 C2"
q: shows "(x >> 1) div y << 1 = word_of_nat (2 * (n div 2 div m))"
B: shows "B = four_block_mat B1 B2 (0\<^sub>m n2 n1) B4"
sink_rev_step_2: fixes q :: "'b"   and q' :: "'b"   and i :: "nat" assumes "\<not> sink q"   and "q = \<delta> q' (w i)" shows "\<not> sink q'"
plus_period: fixes f :: "'a \<Rightarrow> 'b"   and period :: "'a"   and x :: "'a" assumes "periodic_fun_simple f period" shows "f (x + period) = f x"
validCI_not_act: fixes active :: "'a \<Rightarrow> cnf \<Rightarrow> bool"   and cid :: "'a"   and t :: "nat \<Rightarrow> cnf"   and \<gamma> :: "(nat \<Rightarrow> 'b) \<Rightarrow> nat \<Rightarrow> bool"   and tCMP :: "'a \<Rightarrow> cnf \<Rightarrow> 'b"   and t' :: "nat \<Rightarrow> 'b"   and n :: "nat" assumes "\<nexists>i. active cid (t i)"   and "\<gamma> (lnth (dynamic_component.proj tCMP active cid (inf_llist t) @\<^sub>l inf_llist t')) n" shows "dynamic_component.eval tCMP active cid t t' n \<gamma>"
assms: shows "nop_v11 A (1\<^sub>\<nat>) f"
execute_new: fixes n :: "nat"   and x :: "'a"   and h :: "heap" shows "execute (Array.new n x) h = Some (Array.alloc (replicate n x) h)"
simps_1: fixes l :: "'d"   and g :: "'a \<Rightarrow> 'a set"   and p :: "('a, 'b, 'c, 'd) seqp" shows "ctermsl ({l}\<langle>g\<rangle> p) = insert ({l}\<langle>g\<rangle> p) (ctermsl p)"
os_rem_rule: fixes xs :: "'a list"   and b :: "'a node ref option"   and x :: "'a" shows "<os_list xs b> os_rem x b <os_list (removeAll x xs)>\<^sub>t"
singleton_rev_conv: fixes x :: "'d"   and xs :: "'d list" shows "([x] = rev xs) = (xs = [x])"
elims_2: fixes x :: "arctic" assumes "pos_arctic x" obtains n :: "int" where "x = Num_arc n" and "0 \<le> n"
assms: fixes x :: "'a" assumes "x \<in> S" shows "0 \<le> f x"
Id_onI: fixes a :: "'e"   and A :: "'e set" assumes "a \<in> A" shows "(a, a) \<in> Id_on A"
elims_2: fixes x :: "atom set \<times> 'a"   and xa :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and xb :: "'a \<Rightarrow> atom set"   and xc :: "perm"   and xd :: "atom set \<times> 'a" assumes "x \<approx>set xa xb xc xd" obtains bs :: "atom set" and xe :: "'a" and R :: "'a \<Rightarrow> 'a \<Rightarrow> bool" and f :: "'a \<Rightarrow> atom set" and p :: "perm" and cs :: "atom set" and y :: "'a" where "x = (bs, xe)" and "xa = R" and "xb = f" and "xc = p" and "xd = (cs, y)" and "f xe - bs = f y - cs \<and> (f xe - bs) \<sharp>* p \<and> R (p \<bullet> xe) y \<and> p \<bullet> bs = cs"
mult: fixes x :: "'a"   and y :: "'a" assumes "x \<in> A"   and "y \<in> B" shows "x \<otimes> y = y \<otimes> x"
attracted_path: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and W :: "'c set"   and A :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "W \<subseteq> V\<^bsub>G\<^esub>"   and "ParityGame.strategy_attracts G p \<sigma> A W"   and "lset P \<inter> A \<noteq> {}" shows "lset P \<inter> W \<noteq> {}"
strategy_attracts_lset: fixes G :: "('c, 'd) ParityGame_scheme"   and P :: "'c llist"   and v0 :: "'c"   and p :: "Player"   and \<sigma> :: "'c \<Rightarrow> 'c"   and A :: "'c set"   and W :: "'c set" assumes "vmc_path G P v0 p \<sigma>"   and "ParityGame.strategy_attracts G p \<sigma> A W"   and "v0 \<in> A" shows "lset P \<inter> W \<noteq> {}"
low_proc_non0_s\<^sub>L\<^sub>0: shows "low_proc_non0\<^sub>2 s\<^sub>L\<^sub>0"
fv_decomp_rm: fixes A :: "('fun, 'var) extstrand_step list" shows "fv\<^sub>e\<^sub>s\<^sub>t (decomp_rm\<^sub>e\<^sub>s\<^sub>t A) \<subseteq> fv\<^sub>e\<^sub>s\<^sub>t A"
preserves_iso: fixes f :: "'a" assumes "A.iso f" shows "A.iso (A.map f)"
size_4: fixes x21 :: "('c, 'd) tm"   and x22 :: "('c, 'd) tm" shows "size (App x21 x22) = size x21 + size x22 + Suc 0"
bound_of_greater: fixes P :: "'c \<Rightarrow> real"   and x :: "'c" assumes "Expectations.bounded P" shows "P x \<le> bound_of P"
ideal_generated_pair_exists: fixes a :: "'a"   and b :: "'a"   and d :: "'a" assumes "ideal_generated {a, b} = ideal_generated {d}" shows "\<exists>p q. p * a + q * b = d"
surjective: fixes r :: "('f, 'g, 'h, 'i) config_scheme" shows "r = \<lparr>round = round r, state = state r, sent = sent r, rcvd = rcvd r, network = network r, \<dots> = config.more r\<rparr>"
bot: fixes x :: "'a" assumes "\<not> x \<le> bound" shows "bot \<sqsubseteq> f x"
strictly_redundant_def: fixes C :: "'b Literal set"   and S :: "'b Literal set set" shows "strictly_redundant C S = (tautology C \<or> (\<exists>D. D \<in> S \<and> D \<subset> C))"
exhaust_sel: fixes nondetT :: "('d, 'e) nondetT" obtains "nondetT = NondetT (run_nondet nondetT)"
induct: fixes P :: "typ \<Rightarrow> typ \<Rightarrow> ((variable \<times> String.literal set) \<times> typ) list \<Rightarrow> bool"   and a0 :: "typ"   and a1 :: "typ"   and a2 :: "((variable \<times> String.literal set) \<times> typ) list" assumes "\<And>v S T subs. P (Tv v S) T subs"   and "\<And>a Ts b Us subs. (\<And>x xa y xb aa. \<lbrakk>a = b \<and> length Ts = length Us; x \<in> set (zip Ts Us); (xa, y) = x; xb = Some aa\<rbrakk> \<Longrightarrow> P xa y aa) \<Longrightarrow> P (Ty a Ts) (Ty b Us) subs"   and "\<And>v va vb vc subs. P (Ty v va) (Tv vb vc) subs" shows "P a0 a1 a2"
squarefree_decomposition_unique2_1: fixes n :: "nat"   and A2 :: "nat set"   and s2 :: "nat" assumes "0 < n"   and "n = \<Prod>A2 * s2\<^sup>2"   and "\<And>x. x \<in> A2 \<Longrightarrow> prime x"   and "finite A2"   and "0 \<le> s2" shows "A2 = squarefree_part n"
A: fixes u :: "nat \<Rightarrow> 'a" assumes "(\<lambda>n. to_Gromov_completion (u n)) \<longlonglongrightarrow> xi" shows "- ereal (Busemann_function_at xi y x / 2) \<le> liminf (\<lambda>xa. ereal (Gromov_product_at x (u xa) y))"
assms_1: shows "ia \<equiv> if \<forall>m. is_zero_row_upt_k m k A then 0 else mod_type_class.to_nat (GREATEST n. \<not> is_zero_row_upt_k n k A) + 1"
elims_2: fixes x :: "'fsym trm" assumes "wt x" obtains xa :: "Preliminaries.var" where "x = Var xa"   | f :: "'fsym" and Tl :: "'fsym trm list" where "x = Fn f Tl" and "wtFsym f \<and> list_all wt Tl \<and> arOf f = map tpOf Tl"
face: shows "S \<inter> {x. a i \<bullet> x = b i} face_of S"
UNIV: fixes S :: "'a set"   and B :: "'a set set" shows "generated_topology S B S"
map_pmf_comp: fixes f :: "'b \<Rightarrow> 'a"   and g :: "'d \<Rightarrow> 'b"   and M :: "'d pmf" shows "map_pmf f (map_pmf g M) = map_pmf (\<lambda>x. f (g x)) M"
invc2_baldR: fixes l :: "('c \<times> color) tree"   and r :: "('c \<times> color) tree"   and a :: "'c" assumes "invc l"   and "invc2 r" shows "invc2 (baldR l a r)"
surj: shows "\<forall>T. \<exists>S. T = \<lbrakk>S\<rbrakk>"
dom_kildall_def: fixes G :: "nat graph_rec" assumes "cfg_doms G" shows "cfg_doms.dom_kildall G = kildall (fst (snd (cfg_doms.nodes_semi G))) (snd (snd (cfg_doms.nodes_semi G))) (cfg_doms.exec G)"
Cod_in_Obj: fixes f :: "('e set, 'e \<Rightarrow> 'e) concrete_category.arr" assumes "arr f" shows "Cod f \<in> {A. |A| <o \<AA>}"
hyps: shows "finite {a. case a of (a, b) \<Rightarrow> l' = a @ b}"
starfun_def: fixes f :: "'b \<Rightarrow> 'c" shows "*f* f \<equiv> Ifun (star_of f)"
deg_ltrm: fixes R :: "('c, 'd) ring_scheme"   and f :: "nat \<Rightarrow> 'c" assumes "UP_ring R"   and "f \<in> carrier (UP R)" shows "deg R (monom (UP R) (f (deg R f)) (deg R f)) = deg R f"
rbt_lookup_rbt_insert': fixes t :: "('a, 'c) rbt"   and k :: "'a"   and v :: "'c" assumes "rbt_sorted t" shows "rbt_lookup (rbt_insert k v t) = rbt_lookup t(k \<mapsto> v)"
distinct_35: fixes x2 :: "nat"   and x101 :: "nat"   and x102 :: "msg" shows "Number x2 \<noteq> Crypt x101 x102"
is_aligned_get_word_bits: fixes p :: "'b word"   and n :: "nat" assumes "is_aligned p n" obtains "is_aligned p n" and "n < LENGTH('b)"   | "p = 0" and "LENGTH('b) \<le> n"
assms: shows "mi.lowest_tops_impl es bdd = Some (r, bdd')"
it_simr: fixes y :: "'a"   and x :: "'a" assumes "\<lparr>y\<rparr>x\<lparr>x \<cdot> y\<rparr>" shows "\<lparr>y\<rparr>x\<^sup>\<star>\<lparr>x\<^sup>\<star> \<cdot> y\<rparr>"
implies_inf_absorb: fixes x :: "'a"   and y :: "'a" shows "- x \<squnion> x \<sqinter> y = - x \<squnion> y"
cnj_transpose: fixes A :: "(('b, 'd) vec, 'c) vec" shows "(\<chi>i j. cnj_class.cnj (Finite_Cartesian_Product.transpose A $ i $ j)) = Finite_Cartesian_Product.transpose (\<chi>i j. cnj_class.cnj (A $ i $ j))"
IH: fixes ts :: "'a \<Rightarrow> ('b \<times> 'c \<Rightarrow>f nat) option"   and I :: "'a \<Rightarrow> 'd option" assumes "ts_inv_ok ts I"   and "thread_oks ts TAS" shows "I \<subseteq>\<^sub>m upd_invs I P TAS"
ln_ln_pos: fixes x :: "real" assumes "3 \<le> x" shows "0 < ln (ln x)"
simps_2: fixes t :: "'a tree"   and ts :: "'a tree list" shows "height\<^sub>F (t # ts) = max (height t) (height\<^sub>F ts)"
cmb: fixes meets :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and \<I> :: "'b \<Rightarrow> bool"   and e :: "('b \<times> 'b) set"   and m :: "('b \<times> 'b) set"   and b :: "('b \<times> 'b) set"   and ov :: "('b \<times> 'b) set"   and d :: "('b \<times> 'b) set"   and s :: "('b \<times> 'b) set"   and f :: "('b \<times> 'b) set" assumes "class.arelations meets \<I> e m b ov d s f" shows "m O b \<subseteq> b"
path_sub_events: fixes Q :: "'a set" assumes "Q \<in> \<P>" shows "Q \<subseteq> \<E>"
size_12: fixes x51 :: "'b ptrm"   and x52 :: "'b ptrm" shows "size (PPair x51 x52) = size x51 + size x52 + Suc 0"
inter_empty1: fixes i :: "nat_int" shows "i \<sqinter> bot = bot"
sorted_RBT_Set_keys: fixes c :: "'b \<Rightarrow> 'b \<Rightarrow> order"   and rbt :: "'b set_rbt" assumes "ID ccompare = Some c" shows "linorder.sorted (le_of_comp c) (RBT_Set2.keys rbt)"
comp_inv_arr: fixes C :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and f :: "'d"   and g :: "'d" assumes "category C"   and "category.inverse_arrows C f g" shows "C g f = partial_magma.dom C f"
szip_eq: fixes u :: "'b stream"   and v :: "'c stream"   and r :: "'b stream"   and s :: "'c stream" shows "(u ||| v = r ||| s) = (u = r \<and> v = s)"
from_mregex_cong: fixes m :: "nat"   and mr :: "mregex"   and xs :: "Formula.formula list"   and ys :: "Formula.formula list" assumes "ok m mr"   and "\<forall>i<m. xs ! i = ys ! i" shows "from_mregex mr xs = from_mregex mr ys"
i_append_assoc: fixes xs :: "'b list"   and ys :: "'b list"   and f :: "nat \<Rightarrow> 'b" shows "xs \<frown> ys \<frown> f = (xs @ ys) \<frown> f"
set_eq_iff: fixes A :: "'k set"   and B :: "'k set" shows "(A = B) = (\<forall>x. (x \<in> A) = (x \<in> B))"
isometry_onI: fixes X :: "'c set"   and f :: "'c \<Rightarrow> 'd" assumes "\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> dist (f x) (f y) = dist x y" shows "isometry_on X f"
comp_converter_map_converter1: fixes f :: "'a \<Rightarrow> 'g"   and g :: "'h \<Rightarrow> 'b"   and h :: "'i \<Rightarrow> 'e"   and k :: "'f \<Rightarrow> 'j"   and conv :: "('g, 'h, 'i, 'j) converter"   and conv' :: "('e, 'f, 'c, 'd) converter" shows "comp_converter (map_converter f g h k conv) conv' = map_converter f g id id (comp_converter conv (map_converter h k id id conv'))"
HEndPhase0_valueChosen: fixes s :: "state"   and s' :: "state"   and q :: "Proc"   and v :: "InputsOrNi" assumes "HEndPhase0 s s' q"   and "valueChosen s v"   and "v \<in> Inputs"   and "Inv1 s" shows "valueChosen s' v"
indspaceD_range: fixes f :: "('a, 'b) aezfun \<Rightarrow> 'c"   and smult :: "('a, 'b) aezfun \<Rightarrow> 'c \<Rightarrow> 'c"   and G :: "'b set"   and FH :: "('a, 'b) aezfun set"   and V :: "'c set" assumes "f \<in> aezfun_scalar_mult.indspace smult G FH V" shows "range f \<subseteq> V"
2: shows "agree (fmrestrict_set ((vsa \<union> vsb) \<inter> vs) fma) (fmrestrict_set ((vsa \<union> vsb) \<inter> vs) fmb)"
l2_11_b: fixes A :: "'p"   and B :: "'p"   and C :: "'p"   and A' :: "'p"   and B' :: "'p"   and C' :: "'p" assumes "Bet A B C"   and "Bet A' B' C'"   and "Cong A B A' B'"   and "Cong B C B' C'" shows "Cong A C A' C'"