hcju/isabelleps huggingface dataset:
fps_lr_inverse_mult_divring_2: fixes f :: "'a fps"   and g :: "'a fps" shows "fps_right_inverse (f * g) (inverse (fps_nth (f * g) 0)) = fps_right_inverse g (inverse (fps_nth g 0)) * fps_right_inverse f (inverse (fps_nth f 0))"
stern_brocot_iterate: shows "stern_brocot = stern_brocot_iterate"
hashcode_literal_def: fixes s :: "String.literal" shows "hashcode s = hashcode (literal.explode s)"
path2_nontrivial: fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n \<noteq> m" shows "2 \<le> length ns"
sup_ty_opt_def: fixes P :: "'a prog" shows "sup_ty_opt P \<equiv> Err.le (subtype P)"
assms_2: shows "x\<^sup>2 + y\<^sup>2 = 1"
is_natural_2: fixes f :: "'a" assumes "S.arr f" shows "\<ll> (S.cod f) \<cdot>\<^sub>S L f = \<ll> f"
fixp_induct: fixes P :: "('c llist llist \<Rightarrow> 'c llist) \<Rightarrow> bool" assumes "llist.admissible P"   and "P (\<lambda>lconcat. LNil)"   and "\<And>xss. \<lbrakk>P xss; llist.le_fun xss lconcat\<rbrakk> \<Longrightarrow> P (\<lambda>xssa. case xssa of LNil \<Rightarrow> LNil | LCons xs xss' \<Rightarrow> lappend xs (xss xss'))" shows "P lconcat"
conventional': fixes g :: "'g"   and n :: "'node"   and ns :: "'node list"   and m :: "'node"   and v :: "'val"   and v' :: "'val"   and x :: "'node" assumes "g \<turnstile> n-ns\<rightarrow>m"   and "n \<notin> set (tl ns)"   and "v \<in> allDefs g n"   and "v \<in> allUses g m"   and "v' \<in> allDefs g x"   and "var g v' = var g v" shows "x \<notin> set (tl ns)"
constant_function_closed: fixes a :: "'a" assumes "a \<in> carrier R" shows "constant_function (carrier R) a \<in> carrier (function_ring (carrier R) R)"
add_incr_left: fixes b :: "'a"   and a :: "'a" assumes "(0::'a) \<le> b" shows "a \<le> b + a"
divisor_count_squarefree: fixes n :: "nat" assumes "squarefree n"   and "0 < n" shows "divisor_count n = 2 ^ primes_omega n"
simps_1: fixes M :: "('d, 'e) kripke"   and S :: "'d"   and P :: "'e" shows "M,S |=c= Atom P = (P \<in> the (LabelFun M S))"
Comp_Hom_Id: fixes A :: "bool"   and f :: "'arr list"   and B :: "bool" assumes "A \<in> {False, True}"   and "f \<in> Collect (G.path A B)" shows "[] @ f = f"
cr_prod1I: fixes A :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d"   and b :: "'e"   and c' :: "'f" assumes "A a b" shows "cr_prod1 c' A a (b, c')"
Arg_less_0: fixes z :: "complex" shows "(0 \<le> Arg z) = (0 \<le> Im z)"
wf_mbuf2'_def: fixes \<sigma> :: "(char list \<times> 'a list) trace"   and j :: "nat"   and n :: "nat"   and R :: "'a list set"   and \<phi> :: "'a MFOTL.formula"   and \<psi> :: "'a MFOTL.formula"   and buf :: "'a option list set list \<times> 'a option list set list" shows "wf_mbuf2' \<sigma> j n R \<phi> \<psi> buf = wf_mbuf2 (min (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j)) (Monitor.progress \<sigma> \<phi> j) (Monitor.progress \<sigma> \<psi> j) (\<lambda>i. qtable n (fv \<phi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<phi>)) (\<lambda>i. qtable n (fv \<psi>) (mem_restr R) (\<lambda>v. MFOTL.sat \<sigma> (map the v) i \<psi>)) buf"
ideal_pOp_closed: fixes R :: "('e, 'f) Ring_scheme"   and I :: "'e set"   and x :: "'e"   and y :: "'e" assumes "Ring R"   and "ideal R I"   and "x \<in> I"   and "y \<in> I" shows "x \<plusminus>\<^bsub>R\<^esub> y \<in> I"
cring_simprules_16: fixes R :: "('c, 'd) ring_scheme"   and x :: "'c" assumes "cring R"   and "x \<in> carrier R" shows "x \<oplus>\<^bsub>R\<^esub> \<zero>\<^bsub>R\<^esub> = x"
assms_3: shows "h \<turnstile> setter \<rightarrow>\<^sub>h h'"
nth_take: fixes i :: "nat"   and n :: "nat"   and xs :: "'b list" assumes "i < n" shows "take n xs ! i = xs ! i"
poly_mapping_keys_eqI: fixes p :: "'a \<Rightarrow>\<^sub>0 'b"   and q :: "'a \<Rightarrow>\<^sub>0 'b" assumes "keys p = keys q"   and "\<And>t. t \<in> keys p \<Longrightarrow> lookup p t = lookup q t" shows "p = q"
simps_4: fixes uu :: "int"   and v :: "int" shows "hd_coeff1 uu (Le v []) = Le v []"
vec_hom_inj: fixes v :: "real Matrix.vec"   and w :: "real Matrix.vec" assumes "of_real_hom.vec_hom v = of_real_hom.vec_hom w" shows "v = w"
isDERIV_Power: fixes x :: "nat"   and fa :: "floatarith"   and vs :: "real list"   and n :: "nat" assumes "isDERIV x fa vs" shows "isDERIV x (fa ^\<^sub>e n) vs"
dge: shows "derivation_ge (derivation_shift F 0 j) k"
res_lim_pushforward': fixes f :: "(nat \<Rightarrow> int) \<Rightarrow> nat \<Rightarrow> int"   and s :: "nat \<Rightarrow> nat \<Rightarrow> int"   and t :: "nat \<Rightarrow> nat \<Rightarrow> int" assumes "is_Zp_continuous f"   and "is_Zp_cauchy s"   and "t = alt_seq s" shows "res_lim (f \<circ> s) = res_lim (f \<circ> t)"
inv_character_def: fixes \<chi> :: "'c \<Rightarrow> complex" shows "inv_character \<chi> = (\<lambda>a. cnj (\<chi> a))"
Inf_fin_def: fixes inf :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool" assumes "class.semilattice_inf inf less_eq less" shows "semilattice_inf.Inf_fin inf \<equiv> semilattice_set.F inf"
xs: shows "xs \<equiv> as @ pair_relator_halflist s t @ bs"
residue_of_zero_2: fixes k :: "nat" shows "\<zero> k = 0"
cases: fixes P :: "(char list \<times> char list \<times> (char list \<times> ty) list \<times> (char list \<times> ty list \<times> ty \<times> char list list \<times> char list exp) list) list"   and a1 :: "char list exp list"   and a2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and a3 :: "char list exp list"   and a4 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" assumes "P \<turnstile> \<langle>a1,a2\<rangle> [\<Rightarrow>] \<langle>a3,a4\<rangle>" obtains s :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" where "a1 = []" and "a2 = s" and "a3 = []" and "a4 = s"   | e :: "char list exp" and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" and v :: "val" and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" and es :: "char list exp list" and es' :: "char list exp list" and s\<^sub>2 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" where "a1 = e__ # es__" and "a2 = s\<^sub>0__" and "a3 = Val v # es'" and "a4 = s\<^sub>2" and "P \<turnstile> \<langle>e__,s\<^sub>0__\<rangle> \<Rightarrow> \<langle>Val v,s\<^sub>1__\<rangle>" and "P \<turnstile> \<langle>es__,s\<^sub>1__\<rangle> [\<Rightarrow>] \<langle>es',s\<^sub>2\<rangle>"   | e :: "char list exp" and s\<^sub>0 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" and e' :: "char list exp" and s\<^sub>1 :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)" and es :: "char list exp list" where "a1 = e # es" and "a2 = s\<^sub>0" and "a3 = throw e' # es" and "a4 = s\<^sub>1" and "P \<turnstile> \<langle>e,s\<^sub>0\<rangle> \<Rightarrow> \<langle>throw e',s\<^sub>1\<rangle>"
acc_le_listI': fixes r :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and A :: "'b set" assumes "order r"   and "acc A r" shows "acc (\<Union>n. list n A) (Listn.le r)"
simps_2: fixes t\<^sub>1 :: "term"   and t\<^sub>2 :: "term" shows "frees (t\<^sub>1 $ t\<^sub>2) = frees t\<^sub>1 |\<union>| frees t\<^sub>2"
dom_cod: fixes f :: "'a" assumes "hom.arr f" shows "hom.dom (hom.cod f) = hom.cod f"
1: fixes x :: "'a prefix_match option"   and y :: "'a wordinterval"   and x2 :: "'a prefix_match" assumes "\<not> wordinterval_empty rs"   and "(x, y) = wordinterval_CIDR_split1 rs"   and "x = Some x2" shows "wordinterval_to_set y = \<Union> (prefix_to_wordset ` set (wordinterval_CIDR_split_prefixmatch y))"
l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_def: fixes to_tree_order :: "(_) object_ptr \<Rightarrow> ((_) heap, exception, (_) object_ptr list) prog"   and first_in_tree_order :: "(_) object_ptr \<Rightarrow> ((_) object_ptr \<Rightarrow> ((_) heap, exception, 'l option) prog) \<Rightarrow> ((_) heap, exception, 'l option) prog" shows "l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M to_tree_order first_in_tree_order \<equiv> first_in_tree_order = l_first_in_tree_order\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_first_in_tree_order to_tree_order"
neutr: fixes x :: "'c DBMEntry" shows "dbm_add x (Le (0::'c)) = x"
foldD_nest_Un_Int: fixes A :: "'b set"   and C :: "'b set"   and e :: "'a" assumes "finite A"   and "finite C"   and "e \<in> D"   and "A \<subseteq> B"   and "C \<subseteq> B" shows "foldD D (\<cdot>) (foldD D (\<cdot>) e C) A = foldD D (\<cdot>) (foldD D (\<cdot>) e (A \<inter> C)) (A \<union> C)"
sscan_snth: fixes f :: "'d \<Rightarrow> 'b \<Rightarrow> 'b"   and xs :: "'d stream"   and a :: "'b"   and i :: "nat" shows "sscan f xs a !! i = fold f (stake (Suc i) xs) a"
Dp_Mp_eq: fixes f :: "int poly" shows "f = Mp f + Polynomial.smult p (Dp f)"
dim_col_matrix_A: fixes signs :: "rat list list"   and subsets :: "nat list list" shows "dim_col (M_mat signs subsets) = length signs"
eSum_in_Submodule: fixes H :: "'a set"   and K :: "'a set"   and f :: "nat \<Rightarrow> 'a"   and n :: "nat"   and s :: "nat \<Rightarrow> 'b" assumes "H \<subseteq> carrier M"   and "K \<subseteq> H"   and "f \<in> {j. j \<le> n} \<rightarrow> K"   and "s \<in> {j. j \<le> n} \<rightarrow> carrier R" shows "l_comb R (mdl M (fgs R M K)) n s f = l_comb R M n s f"
equiv_up_to_sym: fixes P :: "(char list \<Rightarrow> int) \<Rightarrow> bool"   and c :: "com"   and c' :: "com" shows "(P \<Turnstile> c \<sim> c') = (P \<Turnstile> c' \<sim> c)"
log_main: fixes x :: "int"   and b :: "proper_base"   and y :: "nat"   and "by" :: "int" assumes "0 < x"   and "log_main b x = (y, by)" shows "by = get_base b ^ y \<and> get_base b ^ y \<le> x \<and> x < get_base b ^ Suc y"
tendsto_mult: fixes f :: "'d \<Rightarrow> 'e"   and a :: "'e"   and F :: "'d filter"   and g :: "'d \<Rightarrow> 'e"   and b :: "'e" assumes "(f \<longlongrightarrow> a) F"   and "(g \<longlongrightarrow> b) F" shows "((\<lambda>x. f x * g x) \<longlongrightarrow> a * b) F"
matroid_subset: fixes \<E> :: "'a set" assumes "\<E> \<subseteq> carrier" shows "matroid \<E> (\<E>.indep_in \<E>)"
ordIso_symmetric: fixes r :: "('d \<times> 'd) set"   and r' :: "('e \<times> 'e) set" assumes "r =o r'" shows "r' =o r"
Q_m: fixes P :: "'a \<Rightarrow> bool" assumes "Measurable.pred M P" shows "Measurable.pred M (Q P)"
antisym_conv: fixes less_eq :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and y :: "'d"   and x :: "'d" assumes "class.order less_eq less"   and "less_eq y x" shows "less_eq x y = (x = y)"
bounded_iff: fixes b :: "'c multiset"   and c :: "'c multiset"   and a :: "'c multiset" shows "(b \<union># c \<subseteq># a) = (b \<subseteq># a \<and> c \<subseteq># a)"
Vinv1_inductive_2: shows "{Vinv1} TS.trans v_TS {> Vinv1}"
homotopic_paths_sym_eq: fixes s :: "'b set"   and p :: "real \<Rightarrow> 'b"   and q :: "real \<Rightarrow> 'b" shows "homotopic_paths s p q = homotopic_paths s q p"
IH_2: assumes "x \<in> ifex_var_set (ifex_ite (restrict_top i a False) (restrict_top t a False) (restrict_top e a False))" shows "x \<in> ifex_var_set (restrict_top i a False) \<or> x \<in> ifex_var_set (restrict_top t a False) \<or> x \<in> ifex_var_set (restrict_top e a False)"
IH1_4: fixes j :: "nat" assumes "p - Suc 0 < j"   and "j < length xs'" shows "xs'' ! j = xs' ! j"
prv_imp_neg_imp_neg_imp: fixes \<phi> :: "'fmla"   and \<chi> :: "'fmla" assumes "\<phi> \<in> fmla"   and "\<chi> \<in> fmla"   and "bprv (neg \<phi>)" shows "bprv (imp \<chi> (neg (imp \<chi> \<phi>)))"
fol_equal_def: fixes p :: "fol fm"   and q :: "fol fm" shows "fol_equal p q \<equiv> p = q"
m_eq: fixes x :: "nat \<Rightarrow> val"   and y :: "val" shows "merge (insert 0 (Suc ` V)) (Suc ` V') (x(0 := y), \<rho>) = merge (shift_var_set V) (Suc ` V') (case_nat y (x \<circ> Suc), \<rho>)"
upper_def: fixes interval :: "'a interval" shows "interval.upper interval = (case interval of Interval x1 x2 \<Rightarrow> x2)"
pth_d: fixes x :: "'c" shows "x + (0::'c) \<equiv> x"
pow_carrier_mat: fixes A :: "'a mat"   and n :: "nat"   and k :: "nat" assumes "A \<in> carrier_mat n n" shows "A ^\<^sub>m k \<in> carrier_mat n n"
fact_ge_1: fixes n :: "nat" shows "(1::'a) \<le> fact n"
bre'_cond_abs: fixes \<Sigma> :: "'d set \<times> 'd set \<times> (('d, 'e) ta_rule \<Rightarrow> nat option)"   and Qi :: "'d set" shows "(\<Sigma> \<in> bre'_cond Qi) = (br'_\<alpha> \<Sigma> \<in> bre_cond Qi)"
prod_emb_Diff: fixes M :: "'d set"   and L :: "'d \<Rightarrow> 'e measure"   and K :: "'d set"   and A :: "('d \<Rightarrow> 'e) set"   and B :: "('d \<Rightarrow> 'e) set" shows "prod_emb M L K (A - B) = prod_emb M L K A - prod_emb M L K B"
atLeastAtMost_singleton: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and a :: "'c" assumes "class.order less_eq less" shows "ord.atLeastAtMost less_eq a a = {a}"
\<ff>_step_bound: fixes j :: "nat"   and w :: "'a list" assumes "j \<le> length w" shows "\<ff> w j \<le> \<ff> w (j - 1) + 1"
curry_def: shows "curry = (\<lambda>c x y. c (x, y))"
neg_one_eq_numeral_iff: fixes one :: "'a"   and times :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and n :: "num" assumes "class.ring_char_0 one times plus zero minus uminus" shows "(uminus one = uminus (numeral.numeral one plus n)) = (n = num.One)"
intro: fixes point_set :: "'a set"   and block_collection :: "'a set multiset"   and u_block_size :: "int" assumes "block_design point_set block_collection u_block_size"   and "incomplete_design_axioms point_set u_block_size" shows "incomplete_design point_set block_collection u_block_size"
backwards: assumes "csa \<in> set (find_sgas_aux fss)" shows "\<not> has_no_zeros csa \<and> csa \<in> consistent_sign_vectors_r fss UNIV"
apply_ncrit_connectible: fixes crit :: "nat \<times> 'e \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> bool \<Rightarrow> (bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g \<Rightarrow> bool"   and d :: "'a \<Rightarrow> nat"   and gs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and bs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and hs :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and B :: "(('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) set"   and m :: "nat"   and ps :: "(bool \<times> bool \<times> (('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) \<times> ('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g) list"   and data :: "nat \<times> 'e"   and ic :: "bool"   and q_in_bs :: "bool"   and p :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g"   and q :: "('t \<Rightarrow>\<^sub>0 'f) \<times> nat \<times> 'g" assumes "ncrit_spec crit"   and "dickson_grading d"   and "set gs \<union> set bs \<union> set hs \<subseteq> B"   and "fst ` B \<subseteq> dgrad_p_set d m"   and "snd ` snd ` set ps \<subseteq> set hs \<times> (set gs \<union> set bs \<union> set hs)"   and "unique_idx (gs @ bs @ hs) data"   and "is_Groebner_basis (fst ` set gs)"   and "\<And>p' q'. \<lbrakk>(p', q') \<in> snd ` set (apply_ncrit crit data gs bs hs ps); fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"   and "\<And>p' q'. \<lbrakk>p' \<in> set gs \<union> set bs; q' \<in> set gs \<union> set bs; fst p' \<noteq> 0; fst q' \<noteq> 0\<rbrakk> \<Longrightarrow> crit_pair_cbelow_on d m (fst ` B) (fst p') (fst q')"   and "(ic, q_in_bs, p, q) \<in> set ps"   and "fst p \<noteq> 0"   and "fst q \<noteq> 0"   and "q_in_bs \<Longrightarrow> q \<in> set gs \<union> set bs" shows "crit_pair_cbelow_on d m (fst ` B) (fst p) (fst q)"
except_eq_zeroI: fixes p :: "'b \<Rightarrow>\<^sub>0 'c"   and S :: "'b set" assumes "keys p \<subseteq> S" shows "except p S = 0"
horizontal_homs_axioms: shows "horizontal_homs (\<cdot>) src trg"
pran_res_override: fixes f :: "('a, 'b) pfun"   and g :: "('a, 'b) pfun"   and A :: "'b set" shows "(f + g) \<rhd>\<^sub>p A \<subseteq>\<^sub>p f \<rhd>\<^sub>p A + g \<rhd>\<^sub>p A"
indecomposable_imp_eq: fixes \<alpha> :: "V"   and \<beta> :: "V" assumes "indecomposable \<alpha>"   and "Ord \<beta>"   and "\<beta> < \<alpha>" shows "\<beta> + \<alpha> = \<alpha>"
simple_path_image_uncountable: fixes g :: "real \<Rightarrow> 'b" assumes "simple_path g" shows "uncountable (path_image g)"
induced_automorphism_bij_between_half_chamber_systems_g: shows "bij_betw ((`) \
) (folding_g.\<C> - g \<turnstile> folding_g.\<C>) (g \<turnstile> folding_g.\<C>)"
unfold': fixes E :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> 'res) \<Rightarrow> '\<AA> state \<times> '\<AA> state \<Rightarrow> \<langle>\<langle>bool\<rangle>\<^sub>\<bottom>\<rangle>\<^sub>\<bottom>"   and \<tau> :: "'\<AA> state \<times> '\<AA> state"   and self :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>0"   and a1 :: "('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>1) \<times> ('\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>2)" assumes "cp E"   and "\<tau> \<Turnstile> \<delta> self \<and> (case a1 of (a1, a2) \<Rightarrow> \<lambda>\<tau>. \<tau> \<Turnstile> \<upsilon> a1 \<and> \<tau> \<Turnstile> \<upsilon> a2) \<tau>"   and "\<tau> \<Turnstile> case a1 of (a, b) \<Rightarrow> PRE self a b"   and "\<exists>res. \<tau> \<Turnstile> (case a1 of (a, b) \<Rightarrow> POST self a b) (\<lambda>_. res)"   and "\<And>res. \<tau> \<Turnstile> (case a1 of (a, b) \<Rightarrow> POST self a b) (\<lambda>_. res) \<Longrightarrow> \<tau> \<Turnstile> E (\<lambda>_. res)" shows "\<tau> \<Turnstile> E (case a1 of (a, b) \<Rightarrow> f self a b)"
left_unique_Grp: fixes A :: "'c set"   and f :: "'c \<Rightarrow> 'd" shows "left_unique (BNF_Def.Grp A f) = inj_on f A"
simps_15: fixes x2 :: "'b" shows "set_option (Some x2) = {x2}"
mrconcNone: fixes C :: "'d \<Rightarrow> 'e \<Rightarrow> 'f option"   and x :: "'e"   and p :: "'d list"   and b :: "'d" assumes "applied_rule_rev C x p = \<bottom>" shows "applied_rule_rev C x (b # p) = applied_rule_rev C x [b]"
abs_dif_fiX_bound: fixes C :: "nat set"   and f :: "nat \<Rightarrow> real"   and g :: "nat \<Rightarrow> real"   and x :: "real"   and y :: "real"   and p :: "nat"   and q :: "nat"   and r :: "nat" assumes "\<forall>l\<in>C. \<bar>f l - g l\<bar> \<le> x"   and "\<forall>l\<in>C. \<forall>m\<in>C. \<bar>f l - f m\<bar> \<le> y"   and "p \<in> C"   and "q \<in> C" shows "\<bar>fiX f p r - fiX g q r\<bar> \<le> 2 * \<Delta> + x + y"
del_left_set: fixes t :: "'b tree"   and x :: "'b"   and t' :: "'b tree" assumes "del_left t = (x, t')"   and "t \<noteq> \<langle>\<rangle>" shows "set_tree t = {x} \<union> set_tree t'"
apath_Cons_iff: fixes u :: "'a"   and e :: "'a \<times> 'a"   and es :: "('a \<times> 'a) list"   and w :: "'a" shows "S.apath u (e # es) w = (e \<in> parcs (subdivide G e w) \<and> fst e = u \<and> S.apath (snd e) es w \<and> fst e \<notin> set (pawalk_verts (snd e) es))"
lookup_in_tree: fixes t :: "('b, 'a) RBT.rbt"   and k :: "'b"   and v :: "'a" shows "(RBT.lookup t k = Some v) = ((k, v) \<in> set (RBT.entries t))"
root_in_def: fixes m :: "real"   and M :: "real"   and f :: "real \<Rightarrow> real" shows "root_in m M f = (\<exists>x\<in>{m..M}. f x = 0)"
blockmat_non_empty: shows "\<forall>bs. blockmat bs \<noteq> []"
FL_equivalence_is_L_bisimulation: shows "is_L_bisimulation FL_logically_equivalent"
inf_absorb2: fixes y :: "'a"   and x :: "'a" assumes "y \<le> x" shows "x \<sqinter> y = y"
IH: assumes "\<And>x. x \<in># {#a#} \<times># B \<Longrightarrow> P x = Q (snd x)" shows "filter_mset P ({#a#} \<times># B) = {#a#} \<times># filter_mset Q B"
state_change_on_either_1: fixes i :: "nat"   and t :: "nat" assumes "gap i (Suc t) \<noteq> gap i t" shows "\<not> change_on_neither (block i t) (gap i t)"
simps_1: fixes s :: "'b word"   and e :: "'b word" shows "wordinterval_lowest_element (WordInterval s e) = (if s \<le> e then Some s else None)"
isSnapshot_def: fixes event :: "('d, 'e, 'f) event" shows "isSnapshot event = (\<exists>x4. event = Snapshot x4)"
cindexE_linear_comp: fixes c :: "real"   and lb :: "real"   and ub :: "real"   and f :: "real \<Rightarrow> real"   and b :: "real" assumes "c \<noteq> 0" shows "cindexE lb ub (f \<circ> (\<lambda>x. c * x + b)) = (if 0 < c then cindexE (c * lb + b) (c * ub + b) f else - cindexE (c * ub + b) (c * lb + b) f)"
intros_1: fixes uu :: "typ list"   and uv :: "String.literal"   and T :: "typ" shows "uu \<turnstile>\<^sub>\<tau> Ct uv T : T"
tseytin_update_def: fixes \<A> :: "'a \<Rightarrow> bool"   and F :: "'a formula" shows "tseytin_update \<A> F \<equiv> Let (map (apsnd ((\<Turnstile>) \<A>)) (tseytin_assmt F)) (foldl pair_fun_upd \<A>)"
select_convs_4: fixes inst :: "inst list"   and funcs :: "cl list"   and tab :: "cl option list list"   and mem :: "mem list"   and globs :: "global list"   and more :: "'a" shows "s.mem \<lparr>inst = inst, funcs = funcs, tab = tab, mem = mem, globs = globs, \<dots> = more\<rparr> = mem"
replicate_empty: fixes n :: "nat"   and x :: "'b" shows "(replicate n x = []) = (n = 0)"
rep_eq: fixes x :: "zhmultiset"   and xa :: "zhmultiset" shows "zhmsetmset (x + xa) = zhmsetmset x + zhmsetmset xa"
heap_upd_ok_to_gc_conf: fixes \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and r :: "Vars.var list"   and e :: "exp"   and \<Gamma>'' :: "(Vars.var \<times> exp) list"   and e'' :: "exp"   and S'' :: "stack_elem list" assumes "heap_upds_ok (\<Gamma>, S)"   and "to_gc_conf r (\<Gamma>, e, S) = (\<Gamma>'', e'', S'')" shows "heap_upds_ok (\<Gamma>'', S'')"
subset_not_subset_eq: fixes A :: "'b set"   and B :: "'b set" shows "order.greater B A = (order.greater_eq B A \<and> \<not> order.greater_eq A B)"
consec_un_min: fixes i :: "nat_int"   and j :: "nat_int" shows "consec i j \<longrightarrow> minimum i = minimum (i \<squnion> j)"
base_enabled: fixes vs :: "state \<Rightarrow> 'a"   and c :: "'a"   and F :: "(nat \<Rightarrow> state) \<Rightarrow> bool"   and s :: "nat \<Rightarrow> state" assumes "basevars vs"   and "\<And>u. vs (first u) = c \<Longrightarrow> F (first s ## u)" shows "(Enabled F) s"
next_InvGlobalRecordCount: fixes s :: "('c, 'd) configuration stream" assumes "holds InvGlobalRecordCount s"   and "next s" shows "nxt (holds InvGlobalRecordCount) s"
verit_comp_simplify_3: fixes b' :: "'a"   and a' :: "'a" shows "(\<not> b' \<le> a') = (a' < b')"
simple_match_and_valid: fixes m1 :: "'d simple_match"   and m2 :: "'d simple_match"   and m :: "'d simple_match" assumes "simple_match_valid m1"   and "simple_match_valid m2"   and "simple_match_and m1 m2 = Some m" shows "simple_match_valid m"
intro_nofail_1: fixes S :: "'b nres" shows "(S \<noteq> FAIL) = nofail S"
worec_fixpoint: fixes H :: "('a Bonk_Schramm_extension_unfolded \<Rightarrow> 'b) \<Rightarrow> 'a Bonk_Schramm_extension_unfolded \<Rightarrow> 'b" assumes "adm_wo Bonk_Schramm_extension_unfolded_wo H" shows "worec Bonk_Schramm_extension_unfolded_wo H = H (worec Bonk_Schramm_extension_unfolded_wo H)"