hcju/isabelleps huggingface dataset:
hom_induced_hom: fixes p :: "int"   and X :: "'a topology"   and S :: "'a set"   and Y :: "'b topology"   and T :: "'b set"   and f :: "'a \<Rightarrow> 'b" shows "hom_induced p X S Y T f \<in> hom (relative_homology_group p X S) (relative_homology_group p Y T)"
lm011: fixes R :: "('c \<times> 'd) set"   and x :: "'c" shows "R \<subseteq> R +< {x} \<times> R `` {x}"
simps: fixes xs :: "('a \<times> 'c) list"   and ox :: "'o"   and ys :: "('a \<times> 'c) list \<times> 'o" shows "oalist_eq_raw (xs, ox) ys = (xs = sort_oalist_aux ox ys)"
s: fixes prod :: "'b \<Rightarrow> 'c \<Rightarrow> 'd"   and k :: "enat"   and S :: "'e set"   and f :: "'e \<Rightarrow> 'b"   and g :: "'e \<Rightarrow> 'c" assumes "bounded_bilinear prod"   and "k-smooth_on S f"   and "k-smooth_on S g"   and "open S" shows "k-smooth_on S (\<lambda>x. prod (f x) (g x))"
LC: shows "while brec_cond (brec_step (build_rqrm \<delta>) Qi) (brec_initial Qi \<delta>) = loop"
combine_common_factor: fixes plus :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and times :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and a :: "'c"   and e :: "'c"   and b :: "'c"   and c :: "'c" assumes "class.semiring plus times" shows "plus (times a e) (plus (times b e) c) = plus (times (plus a b) e) c"
bet_neq32__neq: fixes A :: "'p"   and B :: "'p"   and C :: "'p" assumes "Bet A B C"   and "C \<noteq> B" shows "A \<noteq> C"
lconcat_eq_LNil: fixes xss :: "'c llist llist" shows "(lconcat xss = LNil) = (lset xss \<subseteq> {LNil})"
single_cone: fixes x :: "'c" shows "|{x}| =o cone"
log_le_iff: fixes m :: "nat"   and n :: "nat" assumes "m \<le> n" shows "local.log m \<le> local.log n"
arrow: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and D :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and x :: "'d"   and y :: "'c"   and e :: "'d" assumes "arrow_to_functor C D G x y e" shows "partial_magma.ide C y \<and> partial_magma.in_hom D e x (G y)"
elims: fixes x :: "'d DBMEntry"   and y :: "'d" assumes "get_const x = y" obtains c :: "'d" where "x = Le c__" and "y = c__"   | c :: "'d" where "x = Lt c" and "y = c"   | "x = \<infinity>" and "y = undefined"
ltake_Suc_conv_snoc_lnth: fixes m :: "nat"   and xs :: "'b llist" assumes "enat m < llength xs" shows "ltake (enat (Suc m)) xs = ltake (enat m) xs $ <xs ?! m>"
return_True_not_False: fixes b :: "bool" shows "spmf (return_spmf b) True = spmf (return_spmf (\<not> b)) False"
mod_mult_cong_right: fixes c :: "'a"   and a :: "'a"   and b :: "'a"   and d :: "'a" shows "[c mod (a * b) = d] (mod a) = [c = d] (mod a)"
UNIV_3: fixes P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>UNIV. P x) = (\<forall>x. P x)"
sterms_no_call: fixes \<Gamma> :: "'a \<Rightarrow> ('b, 'c, 'a, 'd) seqp"   and pn :: "'a"   and p :: "('b, 'c, 'a, 'd) seqp" assumes "wellformed \<Gamma>" shows "call(pn) \<notin> sterms \<Gamma> p"
add_sum_int: fixes j :: "int"   and k :: "int"   and l :: "int"   and f :: "int \<Rightarrow> 'c" assumes "j < k"   and "k < l" shows "sum f {j..<k} + sum f {k..<l} = sum f {j..<l}"
relevant_endpoint_def: fixes L :: "'b \<Rightarrow> nat"   and V :: "'b set"   and e :: "'b \<times> 'b" shows "relevant_endpoint L V e = (if L (fst e) = 1 then fst e else snd e)"
empty_not_reachable_imp_matchers_empty: fixes P :: "(('e, 'd) Term.term \<times> ('e, 'f) Term.term) multiset"   and \<sigma> :: "'d \<Rightarrow> ('e, 'f) Term.term option" assumes "\<And>\<sigma>''. ((P, \<sigma>), {#}, \<sigma>'') \<notin> matchrel\<^sup>*" shows "matchers_map \<sigma> \<inter> matchers (set_mset P) = {}"
inv_inv: fixes f :: "'c" assumes "C.iso f" shows "C.inv (C.inv f) = f"
tCn: shows "t \<in> SPR.jkbpCn (tLength t)"
lfp_eq_fixp: fixes f :: "'g \<Rightarrow> 'g" assumes "mono f" shows "lfp f = ccpo_class.fixp f"
lsc_at_real: fixes f :: "'b \<Rightarrow> ereal"   and x0 :: "'b" assumes "\<bar>f x0\<bar> \<noteq> \<infinity>" shows "lsc_at x0 f = (\<forall>e>0. \<exists>T. open T \<and> x0 \<in> T \<and> (\<forall>y\<in>T. f x0 - e < f y))"
pcr_cr_eq: shows "pcr_real_int = cr_real_int"
sp_eq_bind_take_drop: fixes F :: "'a \<Rightarrow> 'b fundef option"   and xs :: "('c, 'd, 'a, 'op, 'opinl, 'opubx, 'e, 'f) Ubx.instr list"   and \<Sigma> :: "Unboxed.type option list"   and n :: "nat" shows "Subx.sp F xs \<Sigma> = Subx.sp F (take n xs) \<Sigma> \<bind> Subx.sp F (drop n xs)"
Hermite_of_upt_row_i_Least_ge: fixes i :: "'a"   and A :: "(('b, 'c) vec, 'a) vec"   and k :: "nat"   and ass :: "'b \<Rightarrow> 'b"   and res :: "'b \<Rightarrow> 'b \<Rightarrow> 'b" assumes "\<not> is_zero_row i A"   and "k \<le> mod_type_class.to_nat i" shows "(LEAST n. Hermite_of_upt_row_i A k ass res $ i $ n \<noteq> (0::'b)) = (LEAST n. A $ i $ n \<noteq> (0::'b))"
hinv_r_bal: fixes l :: "'a tree"   and r :: "'a tree"   and n :: "'a" assumes "hinv l"   and "hinv r"   and "height (erase r) = height (erase l) + 2" shows "hinv (r_bal n l r)"
sccs_verts_disjoint: fixes S :: "'a set"   and T :: "'a set" assumes "S \<in> H.sccs_verts"   and "T \<in> H.sccs_verts"   and "S \<noteq> T" shows "S \<inter> T = {}"
Until_eq: shows "\<phi>' = formula.Until \<phi>''' I \<psi>''"
card_of_lists_infinite: fixes A :: "'c set" assumes "infinite A" shows "|lists A| =o |A|"
prems: shows "n'' \<in> sum_SDG_slice2 n"
null_sets_translation: fixes N :: "'b set"   and a :: "'b" assumes "N \<in> null_sets lborel" shows "{x. x - a \<in> N} \<in> null_sets lborel"
sb_def: fixes ys :: "'d"   and y1 :: "'c"   and y2 :: "'c"   and xs :: "'d"   and x :: "'c" shows "x @xs[y1 / y2]_ys \<equiv> if ys = xs then if x = y2 then y1 else x else x"
subclseq_C_D: shows "G\<turnstile>C\<preceq>\<^sub>C D"
simps: fixes n :: "nat"   and as :: "nat"   and lm :: "nat list"   and s :: "nat"   and l :: "cell list"   and r :: "cell list"   and ires :: "cell list" shows "dec_first_on_right_moving n (as, lm) (s, l, r) ires = (\<exists>lm1 lm2 m ml mr rn. lm = lm1 @ [m] @ lm2 \<and> ml + mr = Suc m \<and> length lm1 = n \<and> 0 < ml \<and> 0 < m \<and> (if lm1 = [] then l = Oc \<up> ml @ Bk # Bk # ires else l = Oc \<up> ml @ [Bk] @ <rev lm1> @ Bk # Bk # ires) \<and> (r = Oc \<up> mr @ [Bk] @ <lm2> @ Bk \<up> rn \<or> r = Oc \<up> mr \<and> lm2 = []))"
sf_posI: fixes x :: "nat" assumes "0 < x" shows "0 < sf x"
match_3: fixes g :: "'a"   and f :: "'a"   and h :: "'a" assumes "C.seq g f"   and "C.seq h g" shows "C.seq (h \<cdot> g) f"
c2_2: shows "\<forall>\<^sub>F x in F. R (norm (f2 x)) (c2 * norm (g2 x))"
Texp_Let: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and a :: "Arity" shows "ttree_restr (- domA \<Delta>) (substitute (FBinds \<Delta>\<cdot>(Aheap \<Delta> e\<cdot>a)) (thunks \<Delta>) (Texp e\<cdot>a)) \<sqsubseteq> Texp (Terms.Let \<Delta> e)\<cdot>a"
ListReds1: fixes e :: "char list exp"   and s :: "(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' :: "(nat \<Rightarrow> (char list \<times> (char list \<times> char list \<Rightarrow> val option)) option) \<times> (char list \<Rightarrow> val option)"   and 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 es :: "char list exp list" assumes "P \<turnstile> \<langle>e,s\<rangle> \<rightarrow>* \<langle>e',s'\<rangle>" shows "P \<turnstile> \<langle>e # es,s\<rangle> [\<rightarrow>]* \<langle>e' # es,s'\<rangle>"
Lxy_snoc: fixes xs :: "'b list"   and x :: "'b"   and S :: "'b set" shows "Lxy (xs @ [x]) S = (if x \<in> S then Lxy xs S @ [x] else Lxy xs S)"
abs_eq: fixes xa :: "real_alg_3"   and x :: "real_alg_3" shows "Real_Alg_Quotient xa + Real_Alg_Quotient x = Real_Alg_Quotient (add_3 xa x)"
exhaust_sel: fixes enat :: "enat" obtains "enat = 0"   | "enat = eSuc (epred enat)"
sq_mtx_one_ith_simps_1: fixes i :: "'a" shows "1 $$ i $ i = 1"
init_def: fixes s :: "(nat \<times> form) list" shows "init s = (\<forall>x\<in>set s. fst x = 0)"
assms: shows "action_dom (fst a) (snd a) \<subseteq> vs"
order_zmset_exists_foundation: fixes less_eq :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and less :: "'c \<Rightarrow> 'c \<Rightarrow> bool"   and M :: "'c zmultiset"   and t :: "'c" assumes "class.order less_eq less"   and "0 < zcount M t" shows "\<exists>s. less_eq s t \<and> 0 < zcount M s \<and> (\<forall>u. 0 < zcount M u \<longrightarrow> \<not> less u s)"
B: shows "gauss_jordan_single A \<in> carrier_mat nr nc"
z_def: shows "z \<in> s \<and> (\<forall>y. r\<^sup>+\<^sup>+ y z \<longrightarrow> y \<notin> s)"
distinct_4: fixes x1 :: "'b" shows "p \<noteq> v x1"
writes_small_big: fixes SW :: "('a, 'b, 'c) prog set"   and setter :: "('a, 'b, 'd) prog"   and h :: "'a"   and h' :: "'a"   and P :: "'a \<Rightarrow> 'a \<Rightarrow> bool" assumes "writes SW setter h h'"   and "h \<turnstile> setter \<rightarrow>\<^sub>h h'"   and "\<And>h h' w. \<lbrakk>w \<in> SW; h \<turnstile> w \<rightarrow>\<^sub>h h'\<rbrakk> \<Longrightarrow> P h h'"   and "reflp P"   and "transp P" shows "P h h'"
maxsimpchain_overlap_join: fixes xs :: "'a set list"   and w :: "'a set"   and ys :: "'a set list" assumes "maxsimpchain (xs @ [w])"   and "maxsimpchain (w # ys)" shows "maxsimpchain (xs @ w # ys)"
src_trg: fixes \<mu> :: "'b" assumes "D.arr \<mu>" shows "src\<^sub>D (trg\<^sub>D \<mu>) = trg\<^sub>D \<mu>"
sSplice_envObs: fixes a :: "'a"   and s :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState"   and s' :: "('a, 'ePubAct, 'es, 'pPubAct, 'ps) BEState" assumes "envObs a s = envObs a s'" shows "sSplice a s s' = s"
rel_resumption_is_DoneD: fixes A :: "'f \<Rightarrow> 'g \<Rightarrow> bool"   and B :: "'h \<Rightarrow> 'i \<Rightarrow> bool"   and r1 :: "('f, 'h, 'j) Resumption.resumption"   and r2 :: "('g, 'i, 'j) Resumption.resumption" assumes "Resumption.resumption.rel_resumption A B r1 r2" shows "Resumption.resumption.is_Done r1 = Resumption.resumption.is_Done r2"
fold_commute: fixes xs :: "'a list"   and h :: "'b \<Rightarrow> 'c"   and g :: "'a \<Rightarrow> 'b \<Rightarrow> 'b"   and f :: "'a \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<And>x. x \<in> set xs \<Longrightarrow> (h \<circ>\<circ> g) x = f x \<circ> h" shows "(h \<circ>\<circ>\<circ> fold) g xs = fold f xs \<circ> h"
stream_space_single_set: fixes X :: "nat \<Rightarrow> bool stream \<Rightarrow> 'a"   and r :: "'a stream"   and n :: "nat" assumes "borel_adapt_stoch_proc F X"   and "r \<in> range (proj_stoch_proc X n)" shows "stream_space_single (proj_stoch_proc X n) r \<in> sets (stream_space borel)"
continuous_attains_inf_proper: fixes s :: "'c set"   and a :: "'c"   and f :: "'c \<Rightarrow> 'd"   and r :: "real" assumes "proper s"   and "a \<in> s"   and "continuous_on s f"   and "\<And>z. z \<in> s - cball a r \<Longrightarrow> f a \<le> f z" shows "\<exists>x\<in>s. \<forall>y\<in>s. f x \<le> f y"
iso_lunit': fixes f :: "'a" assumes "B.ide f" shows "B.iso \<l>\<^sup>-\<^sup>1[f]"
elims: fixes x :: "indices list"   and y :: "indices list" assumes "right_app x = y" obtains xa :: "indices list" where "x = xa" and "y = indices.Right # xa"
diff_less: fixes n :: "nat"   and m :: "nat" assumes "0 < n"   and "0 < m" shows "m - n < m"
relcomp_empty1: fixes R :: "('g \<times> 'f) set" shows "{} O R = {}"
E: fixes C :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and \<phi> :: "'c \<times> 'c \<Rightarrow> 'c \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and a :: "'c"   and e :: "'d" assumes "yoneda_lemma_fixed_e C S \<phi> F a e" shows "e \<in> set_category_data.set S (set_category.some_img S (cardSuc (cmax |UNIV| natLeq))) (F a)"
items_idem: fixes H :: "msg set" shows "items (items H) = items H"
Un_assoc: fixes A :: "'x set"   and B :: "'x set"   and C :: "'x set" shows "A \<union> B \<union> C = A \<union> (B \<union> C)"
split_comp_eq_pair: fixes a :: "'b" shows "(\<lambda>x. fst x = a) = (\<lambda>(x, y). x = a)"
states_closed: shows "\<forall>s. set_pmf (\<K> s) \<subseteq> \<S>"
reaches_steps_iff: fixes x :: "'a"   and y :: "'a" shows "(x \<rightarrow>* y) = (\<exists>xs. hd xs = x \<and> last xs = y \<and> local.steps xs)"
circ_sup_one_right_unfold: fixes x :: "'a" assumes "top \<le> x" shows "top \<sqinter> x = top"
natural_transformation_axioms: shows "natural_transformation (\<cdot>\<^sub>A) (\<cdot>\<^sub>B) (A_B.Dom (fst Fg)) (A_B.Dom (fst Fg)) (A_B.Dom (fst Fg))"
new_element_get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t: fixes h :: "(_) heap"   and h' :: "(_) heap"   and ptr :: "(_) document_ptr"   and getter :: "('z, 'x, 'y, 'ag option) RDocument_scheme \<Rightarrow> 'ai" assumes "h \<turnstile> new_element \<rightarrow>\<^sub>h h'" shows "preserved (get_M\<^sub>D\<^sub>o\<^sub>c\<^sub>u\<^sub>m\<^sub>e\<^sub>n\<^sub>t ptr getter) h h'"
weakenL: fixes \<Gamma> :: "'b formula multiset"   and \<Delta> :: "'b formula multiset"   and F :: "'b formula" assumes "\<Gamma> \<Rightarrow> \<Delta>" shows "F, \<Gamma> \<Rightarrow> \<Delta>"
defined_args_valid: fixes \<tau> :: "'\<AA> state \<times> '\<AA> state"   and x :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>"   and y :: "'\<AA> state \<times> '\<AA> state \<Rightarrow> '\<alpha>" shows "(\<tau> \<Turnstile> \<delta> f x y) = (\<tau> \<Turnstile> \<upsilon> x \<and> \<tau> \<Turnstile> \<upsilon> y)"
normalize_mult_normalize_left: fixes a :: "'b poly"   and b :: "'b poly" shows "smult (inverse (lead_coeff (smult (inverse (lead_coeff a)) a * b))) (smult (inverse (lead_coeff a)) a * b) = smult (inverse (lead_coeff (a * b))) (a * b)"
induct: fixes x1 :: "cmsg"   and x2 :: "cmsg"   and P :: "cmsg \<Rightarrow> cmsg \<Rightarrow> bool" assumes "eq x1 x2"   and "\<And>a b c. P (cExp (cExp a b) c) (cExp (cExp a c) b)"   and "\<And>t. P (cTag t) (cTag t)"   and "\<And>A. P (cAgent A) (cAgent A)"   and "\<And>x. P (cNonce x) (cNonce x)"   and "\<And>x. P (cNumber x) (cNumber x)"   and "\<And>x. P (cLtK x) (cLtK x)"   and "\<And>x. P (cEphK x) (cEphK x)"   and "\<And>a b c d. \<lbrakk>eq a b; P a b; eq c d; P c d\<rbrakk> \<Longrightarrow> P (cPair a c) (cPair b d)"   and "\<And>a b c d. \<lbrakk>eq a b; P a b; eq c d; P c d\<rbrakk> \<Longrightarrow> P (cEnc a c) (cEnc b d)"   and "\<And>a b c d. \<lbrakk>eq a b; P a b; eq c d; P c d\<rbrakk> \<Longrightarrow> P (cAenc a c) (cAenc b d)"   and "\<And>a b c d. \<lbrakk>eq a b; P a b; eq c d; P c d\<rbrakk> \<Longrightarrow> P (cSign a c) (cSign b d)"   and "\<And>a b. \<lbrakk>eq a b; P a b\<rbrakk> \<Longrightarrow> P (cHash a) (cHash b)"   and "\<And>a b c d. \<lbrakk>eq a b; P a b; eq c d; P c d\<rbrakk> \<Longrightarrow> P (cExp a c) (cExp b d)"   and "\<And>a b c. \<lbrakk>eq a b; P a b; eq b c; P b c\<rbrakk> \<Longrightarrow> P a c" shows "P x1 x2"
poly_roots_smult: fixes c :: "'a"   and p :: "'a poly" assumes "c \<noteq> (0::'a)" shows "poly_roots (Polynomial.smult c p) = poly_roots p"
KBasic_5: fixes v :: "i"   and \<phi> :: "\<o>"   and \<psi> :: "\<o>" shows "[\<^bold>\<box>(\<phi> \<^bold>\<rightarrow> \<psi>) \<^bold>& \<^bold>\<box>(\<psi> \<^bold>\<rightarrow> \<phi>) \<^bold>\<rightarrow> (\<^bold>\<box>\<phi> \<^bold>\<equiv> \<^bold>\<box>\<psi>) in v]"
simps_1: fixes S :: "nat \<Rightarrow> type_scheme"   and n :: "nat" shows "bound_scheme_inst S (FVar n) = FVar n"
1: shows "[] \<in> {p. valid_path Pi p \<and> distinct p}"
deg_pm_eq_0_iff: fixes s :: "'c \<Rightarrow>\<^sub>0 'b" shows "(deg_pm s = (0::'b)) = (s = 0)"
FactGroup_onto: assumes "h ` carrier G = carrier H" shows "(\<lambda>X. the_elem (h ` X)) ` carrier (G Mod kernel G H h) = carrier H"
compl_le_compl_iff: fixes minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and inf :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and sup :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and bot :: "'a"   and top :: "'a"   and x :: "'a"   and y :: "'a" assumes "class.boolean_algebra minus uminus inf less_eq less sup bot top" shows "less_eq (uminus x) (uminus y) = less_eq y x"
assms_2: shows "kind a = Q\<hookleftarrow>\<^bsub>p\<^esub>f"
assms_1: fixes x :: "Preliminaries.var" shows "\<phi> (Var x)"
entailsI: fixes P :: "'c \<Rightarrow> real"   and Q :: "'c \<Rightarrow> real" assumes "\<And>s. P s \<le> Q s" shows "P \<tturnstile> Q"
run_reachable: fixes r :: "nat \<Rightarrow> 'v" assumes "G'.is_run r" shows "range r \<subseteq> G'.reachable"
set_diff_non_empty_not_subset: fixes A :: "'a set"   and B :: "'a set"   and C :: "'a set" assumes "A \<subseteq> B - C"   and "C \<noteq> {}"   and "A \<noteq> {}"   and "B \<noteq> {}" shows "\<not> A \<subseteq> C"
r_gzero: shows "0 < \<Lambda> * (\<v> - 1) div (\<k> - 1)"
scaleR_left_distrib: fixes a :: "real"   and b :: "real"   and x :: "'c" shows "(a + b) *\<^sub>R x = a *\<^sub>R x + b *\<^sub>R x"
set_rev: fixes xs :: "'f list" shows "set (rev xs) = set xs"
even_of_intvl_intvl: fixes S :: "nat set"   and m :: "nat"   and n :: "nat" assumes "S = {m..n} \<inter> Collect even" shows "\<exists>m' n'. S = (\<lambda>i. i * 2) ` {m'..n'}"
valid_write_sops_def: fixes ts :: "('b \<times> instr list \<times> (nat \<Rightarrow> nat option) \<times> 'b memref list \<times> bool \<times> nat set \<times> (nat \<Rightarrow> bool option)) list" shows "valid_write_sops ts \<equiv> \<forall>i p is \<O> \<R> \<D> \<theta> sb. i < length ts \<longrightarrow> ts ! i = (p, is, \<theta>, sb, \<D>, \<O>, \<R>) \<longrightarrow> (\<forall>sop\<in>write_sops sb. valid_sop sop)"
natural_transformation_def: fixes A :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and B :: "'d \<Rightarrow> 'd \<Rightarrow> 'd"   and F :: "'c \<Rightarrow> 'd"   and G :: "'c \<Rightarrow> 'd"   and \<tau> :: "'c \<Rightarrow> 'd" shows "natural_transformation A B F G \<tau> \<equiv> (category A \<and> category B) \<and> functor A B F \<and> functor A B G \<and> natural_transformation_axioms A B F G \<tau>"
assms_6: shows "is_sig_red (\<prec>\<^sub>t) (=) (set bs) (monom_mult (1::'b) (pp_of_term u - lp a) a)"
ext_inject: fixes Field1 :: "int \<times> day \<Rightarrow> int"   and Field2 :: "int"   and more :: "'a"   and Field1' :: "int \<times> day \<Rightarrow> int"   and Field2' :: "int"   and more' :: "'a" shows "(\<lparr>Field1 = Field1, Field2 = Field2, \<dots> = more\<rparr> = \<lparr>Field1 = Field1', Field2 = Field2', \<dots> = more'\<rparr>) = (Field1 = Field1' \<and> Field2 = Field2' \<and> more = more')"
k_2: shows "i - j cd\<^bsup>\<pi> \<guillemotleft> j\<^esup>\<rightarrow> k"
comp_fun_commute_def: fixes f :: "'b \<Rightarrow> 'c \<Rightarrow> 'c" shows "comp_fun_commute f \<equiv> \<forall>y x. f y \<circ> f x = f x \<circ> f y"
sup_eq_bot_iff: fixes sup :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and less_eq :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and less :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and bot :: "'b"   and x :: "'b"   and y :: "'b" assumes "class.bounded_semilattice_sup_bot sup less_eq less bot" shows "(sup x y = bot) = (x = bot \<and> y = bot)"
infinite_contains_2_elems: fixes A :: "'a set" assumes "infinite A" shows "\<exists>x y. x \<noteq> y \<and> x \<in> A \<and> y \<in> A"
proportionality_def: shows "local.proportionality \<equiv> {(x, y). x \<noteq> (0::'b) \<and> y \<noteq> (0::'b) \<and> (\<exists>k. x = k *s y)}"
local_7: shows "\<S>\<bullet>\<C> \<turnstile> [Local n i vls es] : ts _> ts'"
intTF_def: shows "intTF \<equiv> InfModel.intTF"
\<iota>_in_hom: shows "\<guillemotleft>\<iota> : a \<star> a \<Rightarrow> a\<guillemotright>"
csubspace_is_subspace: fixes A :: "'d set" assumes "csubspace A" shows "subspace A"
mat_of_cols_carrier_1: fixes n :: "nat"   and vs :: "'c vec list" shows "mat_of_cols n vs \<in> carrier_mat n (length vs)"
clausal_Inf_counterex_reducing: fixes N :: "'a literal multiset set"   and D :: "'a literal multiset" assumes "{#} \<notin> N"   and "D \<in> N"   and "\<not> J_of N \<TTurnstile> D"   and "\<And>C. \<lbrakk>C \<in> N; \<not> J_of N \<TTurnstile> C\<rbrakk> \<Longrightarrow> D \<le> C" shows "\<exists>\<iota>\<in>Inf. inference.prems_of \<iota> \<noteq> [] \<and> Standard_Redundancy_Criterion.main_prem_of \<iota> = D \<and> set (Standard_Redundancy_Criterion.side_prems_of \<iota>) \<subseteq> N \<and> J_of N \<TTurnstile>s set (Standard_Redundancy_Criterion.side_prems_of \<iota>) \<and> \<not> J_of N \<TTurnstile> Calculus.inference.concl_of \<iota> \<and> Calculus.inference.concl_of \<iota> < D"
to_list_correct: fixes e :: "'fenv_std" shows "map_of (Fstd_to_list e) = Fstd_get e"
closed_affine_hull: fixes S :: "'a set" shows "closed (affine hull S)"
nat_add_max_right: fixes m :: "nat"   and n :: "nat"   and q :: "nat" shows "m + max n q = max (m + n) (m + q)"
gcd_monic_constant: fixes f :: "'b poly"   and g :: "'b poly" assumes "monic f"   and "degree g = 0" shows "gcd f g \<in> {1, f}"
Q'_sets: fixes i :: "nat" shows "Q' i \<in> sets M"
unx_none_simps_1: fixes name :: "name" shows "unapp (const name) = None"
simps_1: fixes p :: "fm"   and q :: "fm"   and k :: "int" shows "d_\<beta> (And p q) k = (d_\<beta> p k \<and> d_\<beta> q k)"
is_weak_bisimulation_def: fixes R :: "'state \<Rightarrow> 'state \<Rightarrow> bool" shows "S_transform.is_weak_bisimulation R \<equiv> symp R \<and> (\<forall>P Q \<phi>. R P Q \<and> P \<turnstile>\<^sub>S \<phi> \<longrightarrow> (\<exists>Q'. Q \<Rightarrow>\<^sub>S Q' \<and> R P Q' \<and> Q' \<turnstile>\<^sub>S \<phi>)) \<and> (\<forall>P Q. R P Q \<longrightarrow> (\<forall>\<alpha> P'. bn \<alpha> \<sharp>* Q \<longrightarrow> P \<rightarrow>\<^sub>S \<langle>\<alpha>,P'\<rangle> \<longrightarrow> (\<exists>Q'. Q \<Rightarrow>\<langle>\<alpha>\<rangle>\<^sub>S Q' \<and> R P' Q')))"
order_lesseq_imp: fixes x :: "'a"   and y :: "'a" shows "(\<forall>z\<le>x. z \<le> y) = (x \<le> y)"
mem_CD_on_Cd: fixes x :: "X5"   and ds :: "D2 set"   and X :: "X5 set" shows "(x \<in> BSI.CD_on ds X) = (x \<in> BSI.Cd (X5d x) X \<and> X5d x \<in> ds)"
C: shows "\<forall>\<sigma>. mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>t\<^sub>o\<^sub>p \<iota>s ioprog \<sigma> = mbind\<^sub>F\<^sub>a\<^sub>i\<^sub>l\<^sub>S\<^sub>a\<^sub>v\<^sub>e \<iota>s ioprog \<sigma>"
minus_other_side: fixes G :: "('e, 'f) ring_scheme"   and a :: "'e"   and b :: "'e" assumes "abelian_group G"   and "a \<in> carrier G"   and "b \<in> carrier G" shows "(a \<ominus>\<^bsub>G\<^esub> b = \<zero>\<^bsub>G\<^esub>) = (a = b)"
succs_is_node: fixes x :: "bool list" assumes "nfa_is_node A x" shows "list_all (nfa_is_node A) (subset_succs A x)"
e_eq: shows "entropy b T Y = - (LINT (x, y)|(S \<Otimes>\<^sub>M T). Pxy (x, y) * log b (Py y))"
tinstT_trans: fixes ty1 :: "typ"   and ty :: "typ"   and ty2 :: "typ" assumes "tinstT ty1 ty"   and "tinstT ty2 ty1" shows "tinstT ty2 ty"
trans: fixes xs :: "'a list"   and ys :: "'a list"   and zs :: "'a list" assumes "xs <~~> ys"   and "ys <~~> zs" shows "xs <~~> zs"
assms: shows "(A, B, C, D) \<in> hermitean_nonzero"
s_prod_distr: fixes x :: "'a"   and y :: "'a"   and z :: "'a" shows "(x + y) \<cdot> z = x \<cdot> z + y \<cdot> z"
take_bit_int_less_eq_self_iff: fixes n :: "nat"   and k :: "int" shows "(take_bit n k \<le> k) = (0 \<le> k)"
inline1_fixp_induct: fixes P :: "(('h, 'i, 'j) gpv \<Rightarrow> 'k \<Rightarrow> ('h \<times> 'k + 'l \<times> ('j \<times> 'k, 'l, 'm) rpv \<times> ('h, 'i, 'j) rpv) spmf) \<Rightarrow> bool"   and callee :: "'k \<Rightarrow> 'i \<Rightarrow> ('j \<times> 'k, 'l, 'm) gpv" assumes "spmf.admissible (\<lambda>inline1'. P (\<lambda>gpv s. inline1' (gpv, s)))"   and "P (\<lambda>_ _. return_pmf None)"   and "\<And>inline1'. P inline1' \<Longrightarrow> P (\<lambda>gpv s. the_gpv gpv \<bind> case_generat (\<lambda>x. return_spmf (Inl (x, s))) (\<lambda>out rpv. the_gpv (callee s out) \<bind> case_generat (\<lambda>(x, y). inline1' (rpv x) y) (\<lambda>out rpv'. return_spmf (Inr (out, rpv', rpv)))))" shows "P (inline1 callee)"
iso_def: fixes G :: "('f, 'h) monoid_scheme"   and H :: "('g, 'i) monoid_scheme" shows "Group.iso G H = {h \<in> hom G H. bij_betw h (carrier G) (carrier H)}"
simps: fixes ins :: "instr array"   and pc :: "nat"   and s :: "integer" shows "nexts ins (pc, s) = concat (map (\<lambda>(pc, s). map (\<lambda>pc. (pc, s)) (exec' ins s pc)) (nexts1 ins (pc, s)))"