hcju/isabelleps huggingface dataset:
to_gc_conf_eqE: fixes r :: "Vars.var list"   and c :: "(Vars.var \<times> exp) list \<times> exp \<times> stack_elem list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and e :: "exp"   and S :: "stack_elem list" assumes "to_gc_conf r c = (\<Gamma>, e, S)" obtains \<Gamma>' :: "(Vars.var \<times> exp) list" and S' :: "stack_elem list" where "c = (\<Gamma>', e, S')" and "\<Gamma> = restrictA (- set r) \<Gamma>'" and "S = restr_stack (- set r) S' @ map Dummy (rev r)"
let\<^sub>1: fixes \<Delta> :: "(Vars.var \<times> exp) list"   and \<Gamma> :: "(Vars.var \<times> exp) list"   and S :: "stack_elem list"   and e :: "exp" assumes "atom ` domA \<Delta> \<sharp>* \<Gamma>"   and "atom ` domA \<Delta> \<sharp>* S" shows "(\<Gamma>, Terms.Let \<Delta> e, S) \<Rightarrow> (\<Delta> @ \<Gamma>, e, S)"
simps_2: fixes k :: "nat" shows "stirling 0 (Suc k) = 0"
divide_less_floor1: fixes n :: "'a"   and y :: "'a" shows "n / y < of_int \<lfloor>n / y\<rfloor> + (1::'a)"
Mp_f_representative: fixes f :: "int poly" shows "Mp f = to_int_poly (of_int_poly f)"
keep_drop_sels_3: fixes R :: "'a \<Rightarrow> 'b \<Rightarrow> assn" shows "fst (R\<^sup>d) = R"
partition_main_correct: fixes hi :: "nat"   and xs :: "'c list"   and lo :: "nat"   and R :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and h :: "'c \<Rightarrow> 'd" assumes "hi < length xs"   and "lo \<le> hi"   and "\<And>x y z. \<lbrakk>R (h x) (h y); R (h y) (h z)\<rbrakk> \<Longrightarrow> R (h x) (h z)"   and "\<And>x y. R (h x) (h y) \<or> R (h y) (h x)" shows "partition_main R h lo hi xs \<le> SPEC (\<lambda>(xs', p). mset xs = mset xs' \<and> lo \<le> p \<and> p \<le> hi \<and> isPartition_map R h xs' lo hi p \<and> (\<forall>i<lo. xs' ! i = xs ! i) \<and> (\<forall>i. hi < i \<and> i < length xs' \<longrightarrow> xs' ! i = xs ! i))"
hn_refine_cons_post: fixes P :: "assn"   and c :: "'c Heap"   and Q :: "assn"   and R :: "'d \<Rightarrow> 'c \<Rightarrow> assn"   and m :: "'d nres"   and Q' :: "assn" assumes "hn_refine P c Q R m"   and "Q \<Longrightarrow>\<^sub>t Q'" shows "hn_refine P c Q' R m"
s1_1_th1: shows "\<forall>n x y. nat_to_pr n (c_pair x y) = nat_to_pr (s1_1 n x) y"
in_cong: fixes f :: "'a \<Rightarrow> 'b"   and g :: "'a \<Rightarrow> 'b"   and F :: "'a filter"   and h :: "'a \<Rightarrow> 'b" assumes "\<forall>\<^sub>F x in F. f x = g x" shows "(f \<in> \<omega>[F](h)) = (g \<in> \<omega>[F](h))"
termFSwSbImorph_iff: fixes h :: "('index, 'bindex, 'varSort, 'var, 'opSym) qTerm set \<Rightarrow> 'g"   and hA :: "('index, 'bindex, 'varSort, 'var, 'opSym) qAbs set \<Rightarrow> 'h"   and MOD :: "('index, 'bindex, 'varSort, 'sort, 'opSym, 'var, 'g, 'h, 'i) model_scheme" shows "termFSwSbImorph h hA MOD = (termFSbImorph h hA MOD \<and> ipresSwapAll h hA MOD)"
sets_C: fixes x :: "nat \<Rightarrow> 'a"   and n :: "nat"   and m :: "nat" assumes "x \<in> space (Pi\<^sub>M {0..<n} (\<lambda>_. M))" shows "sets (C n m x) = sets (Pi\<^sub>M {0..<n + m} (\<lambda>_. M))"
formulaEntailsLiteralAppend: fixes formula :: "Literal list list"   and literal :: "Literal"   and formula' :: "Literal list list" assumes "formulaEntailsLiteral formula literal" shows "formulaEntailsLiteral (formula @ formula') literal"
set_zip: fixes xs :: "'k list"   and ys :: "'l list" shows "set (zip xs ys) = {(xs ! i, ys ! i) |i. i < min (length xs) (length ys)}"
map_eq: shows "\<forall>k\<in>set (map fst xs). map_of xs k = map_of ys k"
abs_eq: fixes x :: "'a" shows "norm (to_conjugate_space x) = norm x"
cil_commute: fixes w :: "'d list"   and \<alpha> :: "'d \<Rightarrow> 'e set \<times> 'e set"   and w' :: "'d list" shows "w \<otimes>\<^bsub>\<alpha>\<^esub> w' = w' \<otimes>\<^bsub>\<alpha>\<^esub> w"
rng: shows "range (\<lambda>f. poly_eval (\<lambda>x. monomial (a x) 0) (focus X f)) = P[- X]"
convr_invol: fixes p :: "('b, 'c \<times> 'd) uexpr" shows "p\<^sup>-\<^sup>- = p"
set_iterator_image_correct: fixes it :: "('c \<Rightarrow> bool) \<Rightarrow> ('d \<Rightarrow> 'c \<Rightarrow> 'c) \<Rightarrow> 'c \<Rightarrow> 'c"   and S :: "'d set"   and g :: "'d \<Rightarrow> 'e"   and S' :: "'e set" assumes "set_iterator it S"   and "inj_on g S"   and "S' = g ` S" shows "set_iterator (set_iterator_image g it) S'"
assms_1: fixes t :: "real" assumes "t \<in> {0..1}" shows "x0 + t *\<^sub>R a \<in> G"
ball_Un: fixes A :: "'a set"   and B :: "'a set"   and P :: "'a \<Rightarrow> bool" shows "(\<forall>x\<in>A \<union> B. P x) = ((\<forall>x\<in>A. P x) \<and> (\<forall>x\<in>B. P x))"
le_cong: fixes x :: "'a"   and y :: "'a"   and z :: "'a"   and w :: "'a" assumes "x .= y"   and "z .= w"   and "x \<in> carrier L"   and "y \<in> carrier L"   and "z \<in> carrier L"   and "w \<in> carrier L" shows "(x \<sqsubseteq> z) = (y \<sqsubseteq> w)"
eval_Fields_conv: fixes P :: "'a prog"   and C :: "String.literal" shows "pred.eval (Fields_i_i_o P C) = Fields P C"
ldropn_lmap: fixes n :: "nat"   and f :: "'c \<Rightarrow> 'b"   and xs :: "'c llist" shows "ldropn n (lmap f xs) = lmap f (ldropn n xs)"
timpls_transformable_to'_if_term_variants: fixes a :: "'g set"   and b :: "'g set"   and s :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term"   and TI :: "('g set \<times> 'g set) list" assumes "term_variants_pred ((\<lambda>_. [])(Abs a := [Abs b])) s t"   and "(a, b) \<in> (set TI)\<^sup>+" shows "timpls_transformable_to' TI s t"
inject: fixes x :: "nat"   and ya :: "nat" shows "(Friend x = Friend ya) = (x = ya)"
hitting_Q_or_new_last_def: fixes G :: "('c, 'd) Graph_scheme"   and v0 :: "'c"   and v1 :: "'c"   and paths :: "'c list set"   and P_new :: "'c list"   and sep_size :: "nat" assumes "ProofStepInduct_NonTrivial G v0 v1 paths P_new sep_size" shows "ProofStepInduct_NonTrivial.hitting_Q_or_new_last G v0 v1 paths P_new sep_size \<equiv> \<lambda>y. y \<noteq> v0 \<and> (y = DisjointPathsPlusOne.new_last P_new \<or> (\<exists>Q_hit\<in>ProofStepInduct_NonTrivial.Q G v0 v1 paths P_new sep_size. y \<in> set Q_hit))"
xor_bot: fixes x :: "'e" assumes "x \<in> carrier L" shows "x \<oplus> \<bottom> = x"
subst_typ'_subst_bv1: fixes insts :: "((variable \<times> String.literal set) \<times> typ) list"   and t :: "Core.term"   and n :: "nat"   and u :: "Core.term" shows "subst_typ' insts (subst_bv1 t n u) = subst_bv1 (subst_typ' insts t) n (subst_typ' insts u)"
lifted_abs_def: shows "lifted_abs \<equiv> PRJ('a\<cdot>lifted) oo EMB('a\<^sub>\<bottom>)"
p: shows "extended_to_set p \<subseteq> I ` set s"
has_integral_component_le: fixes k :: "'a"   and f :: "'b \<Rightarrow> 'a"   and i :: "'a"   and S :: "'b set"   and g :: "'b \<Rightarrow> 'a"   and j :: "'a" assumes "k \<in> Basis"   and "(f has_integral i) S"   and "(g has_integral j) S"   and "\<And>x. x \<in> S \<Longrightarrow> f x \<bullet> k \<le> g x \<bullet> k" shows "i \<bullet> k \<le> j \<bullet> k"
5: shows "(q \<otimes>\<^bsub>P\<^esub> p) (deg R p + deg R q) = (monom P (q (deg R q)) (deg R q) \<otimes>\<^bsub>P\<^esub> monom P (p (deg R p)) (deg R p)) (deg R p + deg R q)"
graph_empty_e: fixes v :: "'c set" shows "graph (LG {} v)"
sig_trd_aux_red_rtrancl: fixes args0 :: "'a \<times> ('t \<Rightarrow>\<^sub>0 'b)" assumes "fst args0 \<in> keys (rep_list (snd args0))" shows "(sig_red (\<prec>\<^sub>t) (\<preceq>) (set bs))\<^sup>*\<^sup>* (snd args0) (sig_trd_aux args0)"
simps: fixes xs :: "nat list" shows "godel_code xs = (let lh = length xs in 2 ^ lh * godel_code' xs (Suc 0))"
hequiv_refl_rel: fixes A :: "'c set"   and H :: "(('d, 'c) fm list \<times> 'c) set" assumes "Hintikka A H" shows "refl_on (names H \<inter> A) (hequiv_rel A H)"
open_unresolvable_def: fixes X :: "'a set" shows "open_unresolvable X = (K \<circ> I \<circ> K = K \<circ> I)"
term_ok'_subst_bv: fixes \<Theta> :: "((String.literal \<Rightarrow> typ option) \<times> (String.literal \<Rightarrow> nat option) \<times> (String.literal \<times> String.literal) set \<times> (String.literal \<Rightarrow> (String.literal \<Rightarrow> String.literal set list option) option)) \<times> Core.term set"   and f :: "Core.term"   and u :: "Core.term" assumes "term_ok' (sig \<Theta>) f"   and "term_ok' (sig \<Theta>) u" shows "term_ok' (sig \<Theta>) (subst_bv f u)"
measure_bind: fixes f :: "'c \<Rightarrow> 'd measure"   and p :: "'c spmf"   and N :: "'d measure"   and X :: "'d set" assumes "f \<in> measure_spmf p \<rightarrow>\<^sub>M subprob_algebra N"   and "X \<in> sets N" shows "Sigma_Algebra.measure (measure_spmf p \<bind> f) X = LINT x|measure_spmf p. Sigma_Algebra.measure (f x) X"
remove_vertices_v0_v1_Digraph: fixes v :: "'a" assumes "v \<noteq> v0"   and "v \<noteq> v1" shows "v0_v1_Digraph (remove_vertex v) v0 v1"
triv_finprod_imp_compl_fam: fixes I :: "'d set"   and S :: "'d \<Rightarrow> 'a set" assumes "finite I"   and "\<And>i. i \<in> I \<Longrightarrow> subgroup (S i) G"   and "\<forall>f\<in>Pi I S. finprod G f I = \<one> \<longrightarrow> (\<forall>i\<in>I. f i = \<one>)" shows "compl_fam S I"
mk_circline_cmat_def: fixes A :: "complex"   and B :: "complex"   and C :: "complex"   and D :: "complex" shows "mk_circline_cmat A B C D = (let M = (A, B, C, D) in if M \<in> hermitean_nonzero then M else eye)"
lookup_cache_usesD: fixes g :: "'g"   and n :: "'node"   and vs :: "'val set" assumes "Mapping.lookup (cache g (uses g)) n = Some vs" shows "vs = uses g n"
rtrancl_into_trancl2: fixes a :: "'b"   and b :: "'b"   and r :: "('b \<times> 'b) set"   and c :: "'b" assumes "(a, b) \<in> r"   and "(b, c) \<in> r\<^sup>*" shows "(a, c) \<in> trancl r"
elims: fixes x :: "'b eclause"   and y :: "'b trm set" assumes "trms_ecl x = y" obtains C :: "'b literal set" and X :: "'b trm set" where "x = Ecl C X" and "y = X"
has_integral_twiddle: fixes r :: "real"   and h :: "'d \<Rightarrow> 'c"   and g :: "'c \<Rightarrow> 'd"   and f :: "'d \<Rightarrow> 'e"   and i :: "'e"   and a :: "'d"   and b :: "'d" assumes "0 < r"   and "\<And>x. h (g x) = x"   and "\<And>x. g (h x) = x"   and "\<And>x. isCont g x"   and "\<And>u v. \<exists>w z. g ` cbox u v = cbox w z"   and "\<And>u v. \<exists>w z. h ` cbox u v = cbox w z"   and "\<And>u v. content (g ` cbox u v) = r * content (cbox u v)"   and "(f has_integral i) (cbox a b)" shows "((\<lambda>x. f (g x)) has_integral (1 / r) *\<^sub>R i) (h ` cbox a b)"
integral_nonneg: fixes M :: "'a measure"   and f :: "'a \<Rightarrow> real" assumes "\<And>x. x \<in> space M \<Longrightarrow> 0 \<le> f x" shows "0 \<le> integral\<^sup>L M f"
Hausdorff_space_compact_sets: fixes X :: "'c topology" shows "Hausdorff_space X = (\<forall>S T. compactin X S \<and> compactin X T \<and> disjnt S T \<longrightarrow> (\<exists>U V. openin X U \<and> openin X V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> disjnt U V))"
boundedByIncLevel': fixes N :: "nat"   and A :: "(nat \<times> 'b) set" shows "boundedBy (Suc N) (incLevel ` A) = boundedBy N A"
is_sorted_list_of_set_by_def: fixes R :: "('b \<times> 'b) set"   and A :: "'b set"   and xs :: "'b list" shows "is_sorted_list_of_set_by R A xs = ((\<forall>i<length xs - 1. (xs ! i, xs ! (i + 1)) \<in> R) \<and> ran\<^sub>l xs = A)"
N_dvd_mono: fixes q :: "real"   and m :: "int"   and n :: "int" assumes "fbnorm q"   and "m dvd n" shows "fbnorm.N q n \<le> fbnorm.N q m"
fds_ind_False: shows "fds_ind (\<lambda>_. False) = 0"
assms_6: shows "mix_pmf \<alpha> p r \<succeq>[P] mix_pmf \<alpha> q r"
outstanding_refs_is_non_volatile_Write\<^sub>s\<^sub>b_takeWhile_conv: fixes sb :: "'b memref list" shows "outstanding_refs is_non_volatile_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb) = outstanding_refs is_Write\<^sub>s\<^sub>b (takeWhile (Not \<circ> is_volatile_Write\<^sub>s\<^sub>b) sb)"
x\<^sub>0'_min_def: shows "x\<^sub>0'_min = max A (max X (max a0 (max (real gx1) (max 1 (real x\<^sub>1 + 1)))))"
arrow: shows "ide x \<and> \<guillemotleft>e : local.prod x b \<rightarrow> c\<guillemotright>"
hull_hull: fixes S :: "'b set \<Rightarrow> bool"   and s :: "'b set" shows "S hull (S hull s) = S hull s"
is_tangle_make_vert_left: fixes xs :: "wall"   and n :: "nat" assumes "is_tangle_diagram xs" shows "is_tangle_diagram (basic (make_vert_block n) \<otimes> xs)"
Goal: fixes \<Psi>' :: "'b"   and \<Psi>\<^sub>Q' :: "'b"   and Pa :: "('a, 'b, 'c) psi"   and Ma :: "'a"   and xveca :: "name list"   and Na :: "'a"   and P'a :: "('a, 'b, 'c) psi"   and A\<^sub>P' :: "name list"   and \<Psi>\<^sub>P' :: "'b"   and Qa :: "('a, 'b, 'c) psi"   and Ka :: "'a"   and Q'a :: "('a, 'b, 'c) psi"   and A\<^sub>Q' :: "name list" assumes "\<Psi>' \<otimes> \<Psi>\<^sub>Q' \<rhd> Pa \<longmapsto> Ma\<lparr>\<nu>*xveca\<rparr>\<langle>Na\<rangle> \<prec> P'a"   and "extractFrame Pa = \<langle>A\<^sub>P', \<Psi>\<^sub>P'\<rangle>"   and "distinct A\<^sub>P'"   and "\<Psi>' \<otimes> \<Psi>\<^sub>P' \<rhd> Qa \<longmapsto> Ka\<lparr>Na\<rparr> \<prec> Q'a"   and "extractFrame Qa = \<langle>A\<^sub>Q', \<Psi>\<^sub>Q'\<rangle>"   and "distinct A\<^sub>Q'"   and "\<Psi>' \<otimes> \<Psi>\<^sub>P' \<otimes> \<Psi>\<^sub>Q' \<turnstile> Ma \<leftrightarrow> Ka"   and "A\<^sub>P' \<sharp>* \<Psi>'"   and "A\<^sub>P' \<sharp>* Pa"   and "A\<^sub>P' \<sharp>* Qa"   and "A\<^sub>P' \<sharp>* Ma"   and "A\<^sub>P' \<sharp>* A\<^sub>Q'"   and "A\<^sub>Q' \<sharp>* \<Psi>'"   and "A\<^sub>Q' \<sharp>* Pa"   and "A\<^sub>Q' \<sharp>* Qa"   and "A\<^sub>Q' \<sharp>* Ka"   and "xveca \<sharp>* Qa" shows "\<Psi>' \<rhd> Pa \<parallel> Qa \<longmapsto> \<tau> \<prec> \<lparr>\<nu>*xveca\<rparr>P'a \<parallel> Q'a"
ide_compE: fixes g :: "'b"   and f :: "'b" assumes "J1.ide (J1 g f)" obtains "J1.seq g f" and "J1.seq f g" and "J1 g f = J1.dom f" and "J1 g f = J1.cod g"
iprev_nth_singleton: fixes a :: "nat"   and n :: "nat" shows "{a} \<leftarrow> n = a"
exposed_face_of_Int_supporting_hyperplane_le: fixes S :: "'b set"   and a :: "'b"   and b :: "real" assumes "convex S"   and "\<And>x. x \<in> S \<Longrightarrow> a \<bullet> x \<le> b" shows "S \<inter> {x. a \<bullet> x = b} exposed_face_of S"
fsqrt_def: fixes m :: "roundmode"   and a :: "('c, 'd) IEEE.float" shows "fsqrt m a = (if is_nan a then some_nan else if is_zero a \<or> is_infinity a \<and> sign a = 0 then a else if sign a = 1 then some_nan else zerosign (sign a) (IEEE.round m (sqrt (valof a))))"
type_preservation: fixes t :: "lam"   and t' :: "lam"   and \<Gamma> :: "(name \<times> ty) list"   and T :: "ty" assumes " t \<longrightarrow>\<^sub>\<beta> t'"   and "\<Gamma> \<turnstile> t : T" shows "\<Gamma> \<turnstile> t' : T"
reachable_def: fixes s :: "nat"   and e :: "((nat \<times> nat) \<times> transition) fset" shows "reachable s e = (\<exists>t. visits s e 0 <> t)"
timpl_apply_term_def: fixes a :: "'g set"   and b :: "'g set"   and t :: "(('e, 'f, 'g) prot_fun, 'h) Term.term" shows "\<langle>a --\<guillemotright> b\<rangle>\<langle>t\<rangle> \<equiv> term_variants ((\<lambda>_. [])(Abs a := [Abs b])) t"
representation_scale: fixes basis :: "'b set"   and v :: "'b"   and r :: "complex" assumes "cindependent basis"   and "v \<in> cspan basis" shows "crepresentation basis (r *\<^sub>C v) = (\<lambda>b. r * crepresentation basis v b)"
image_mset_subst_outside: fixes x :: "'b"   and s :: "'b multiset"   and f :: "'b \<Rightarrow> 'c"   and y :: "'c" assumes "x \<notin># s" shows "image_mset (f(x := y)) s = image_mset f s"
sup_top_left: fixes x :: "'a" shows "\<top> \<squnion> x = \<top>"
Cd_singleton: fixes x :: "'a"   and d :: "'b"   and X :: "'a set" shows "(x \<in> Pdds'.Cd d X) = (Pdds'.Cd d X = {x})"
wf_operator_preD: fixes name :: "char list"   and pres :: "(nat \<times> nat) list"   and effs :: "((nat \<times> nat) list \<times> nat \<times> nat option \<times> nat) list"   and cost :: "nat" assumes "wf_operator (name, pres, effs, cost)" shows "wf_partial_state pres"
ide_mkObj: fixes a :: "'a" assumes "C.ide a" shows "V.ide (mkObj a)"
ih2: shows "pi \<bullet> (\<parallel><A>\<parallel>) = \<parallel><A>\<parallel>"
axioms_1: fixes V\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and H\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<a>\<^sub>C :: "'b \<Rightarrow> 'b \<Rightarrow> 'b \<Rightarrow> 'b"   and \<i>\<^sub>C :: "'b \<Rightarrow> 'b"   and src\<^sub>C :: "'b \<Rightarrow> 'b"   and trg\<^sub>C :: "'b \<Rightarrow> 'b"   and V\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and H\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<a>\<^sub>D :: "'c \<Rightarrow> 'c \<Rightarrow> 'c \<Rightarrow> 'c"   and \<i>\<^sub>D :: "'c \<Rightarrow> 'c"   and src\<^sub>D :: "'c \<Rightarrow> 'c"   and trg\<^sub>D :: "'c \<Rightarrow> 'c"   and F :: "'b \<Rightarrow> 'c"   and \<Phi> :: "'b \<times> 'b \<Rightarrow> 'c" assumes "equivalence_pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>" shows "pseudofunctor V\<^sub>C H\<^sub>C \<a>\<^sub>C \<i>\<^sub>C src\<^sub>C trg\<^sub>C V\<^sub>D H\<^sub>D \<a>\<^sub>D \<i>\<^sub>D src\<^sub>D trg\<^sub>D F \<Phi>"
hyps: fixes X :: "nat set" assumes "a \<in> unforwarded_non_volatile_reads sb X" shows "\<exists>xs ys t v. sb = xs @ Read\<^sub>s\<^sub>b False a t v # ys \<and> a \<notin> outstanding_refs is_Write\<^sub>s\<^sub>b xs"
elims: fixes x :: "nat"   and xa :: "nat"   and y :: "'b" assumes "token_run x xa = y" obtains xb :: "nat" and n :: "nat" where "x = xb" and "xa = n" and "y = run \<delta> q\<^sub>0 (suffix xb w) (n - xb)"
normalize_mult_unit_right: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and unit_factor :: "'b \<Rightarrow> 'b"   and normalize :: "'b \<Rightarrow> 'b"   and b :: "'b"   and a :: "'b" assumes "class.normalization_semidom divide plus minus zero times one unit_factor normalize"   and "dvd.dvd times b one" shows "normalize (times a b) = normalize a"
otherwithI: fixes I :: "'d set"   and Q :: "'e \<Rightarrow> 'e \<Rightarrow> bool"   and \<sigma> :: "'d \<Rightarrow> 'e"   and \<sigma>' :: "'d \<Rightarrow> 'e"   and P :: "('d \<Rightarrow> 'e) \<Rightarrow> 'f \<Rightarrow> bool"   and a :: "'f" assumes "\<And>j. j \<notin> I \<Longrightarrow> Q (\<sigma> j) (\<sigma>' j)"   and "P \<sigma> a" shows "otherwith Q I P \<sigma> \<sigma>' a"
pivot_funD_5: fixes A :: "'c mat"   and nr :: "nat"   and f :: "nat \<Rightarrow> nat"   and nc :: "nat"   and i :: "nat"   and i' :: "nat" assumes "dim_row A = nr"   and "pivot_fun A f nc"   and "i < nr"   and "f i < nc"   and "i' < nr"   and "i' \<noteq> i" shows "A $$ (i', f i) = (0::'c)"
env_C_restr_def: shows "env_C_restr = (\<Lambda> r f. cfun_comp\<cdot>(C_restr\<cdot>r)\<cdot>f)"
r_e2tail_prim: shows "prim_recfn 1 r_e2tail"
merge_on_subset: fixes B :: "'a set" assumes "B \<subseteq> UNIV" shows "merge_on B rha boa ma"
span_superset: fixes S :: "('c \<Rightarrow>\<^sub>0 'd) set" shows "S \<subseteq> phull S"
ladder_i_in_\<alpha>: shows "ladder_i L 0 < length \<alpha>"
unique_poincare_line: fixes u :: "complex_homo"   and v :: "complex_homo"   and l :: "circline" assumes "u \<noteq> v"   and "u \<in> unit_disc"   and "v \<in> unit_disc"   and "u \<in> circline_set l"   and "v \<in> circline_set l"   and "is_poincare_line l" shows "l = poincare_line u v"
is_extensional: fixes f :: "'t" assumes "\<not> S'.arr f" shows "\<Psi> f = S.null"
sets_measure_pmf_count_space: fixes M :: "'c pmf" shows "measure_pmf.events M = sets (count_space UNIV)"
component_is_regular: fixes e :: "'a"   and v :: "'a" shows "regular (choose_component e v)"
refines_antisym: fixes x :: "'a"   and y :: "'a" assumes "refines x y"   and "refines y x" shows "x = y"
cond_def: fixes P :: "'c \<times> 'c \<Rightarrow> bool"   and b :: "'c \<times> 'c \<Rightarrow> bool"   and Q :: "'c \<times> 'c \<Rightarrow> bool" shows "P \<triangleleft> b \<triangleright>  Q \<equiv> b \<and> P \<or> \<not> b \<and> Q"
eq_fst_iff: fixes a :: "'d"   and p :: "'d \<times> 'e" shows "(a = fst p) = (\<exists>b. p = (a, b))"
Sup_le_iff: fixes A :: "'al set"   and b :: "'al" assumes "A \<subseteq> U"   and "b \<in> U" shows "(\<Squnion>\<^sub>o\<^sub>w A \<le>\<^sub>o\<^sub>w b) = (\<forall>x\<in>A. x \<le>\<^sub>o\<^sub>w b)"
rtrancl_while_Some: fixes p :: "'c \<Rightarrow> bool"   and f :: "'c \<Rightarrow> 'c list"   and x :: "'c"   and ws :: "'c list"   and Z :: "'c set" assumes "rtrancl_while p f x = Some (ws, Z)" shows "if ws = [] then Z = {(x, y). y \<in> set (f x)}\<^sup>* `` {x} \<and> (\<forall>z\<in>Z. p z) else \<not> p (hd ws) \<and> hd ws \<in> {(x, y). y \<in> set (f x)}\<^sup>* `` {x}"
nonintS_iff_nonint: fixes st0 :: "'a"   and do :: "'a \<Rightarrow> 'b \<Rightarrow> 'c \<Rightarrow> 'a"   and out :: "'a \<Rightarrow> 'b \<Rightarrow> 'd"   and GH :: "'b set"   and GL :: "'b set" shows "GM_sec_model.nonintS st0 do out GH GL = GM_sec_model.nonint st0 do out GH GL"
simps: fixes dims :: "nat list"   and vars :: "nat set"   and i :: "nat"   and j :: "nat" shows "partial_state.encode12 dims vars (i, j) = digit_decode dims (weave vars (digit_encode (partial_state.dims1 dims vars) i) (digit_encode (partial_state.dims2 dims vars) j))"
fpxs_compose_power_1_right: fixes f :: "'b fpxs" shows "fpxs_compose_power f 1 = f"
limes_def: fixes T :: "'b set set"   and F :: "'b set set" shows "lim\<^bsub>T\<^esub> F = (THE x. x \<in> carrier\<^bsub>T\<^esub> \<and> F \<longlongrightarrow>\<^bsub>T\<^esub> x)"
tauCongChainBangI: fixes P :: "('a, 'b, 'c) psi"   and P' :: "('a, 'b, 'c) psi"   and \<Psi> :: "'b" assumes "\<Psi> \<rhd> P \<parallel> P \<Longrightarrow>\<^sub>\<tau> P'"   and "guarded P" obtains Q :: "('a, 'b, 'c) psi" where "\<Psi> \<rhd> !P \<Longrightarrow>\<^sub>\<tau> Q" and "\<Psi> \<rhd> Q \<sim> P' \<parallel> !P"
chine_in_hom: shows "\<guillemotleft>\<mu>.chine : \<mu>.dom.apex \<rightarrow>\<^sub>C \<mu>.cod.apex\<guillemotright>"
list_set_tl: fixes x :: "'a"   and xs :: "'a list" assumes "x \<in> set (tl xs)" shows "x \<in> set xs"
intro: fixes le :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and lub :: "(nat \<Rightarrow> 'a) \<Rightarrow> 'a" assumes "porder le"   and "omega_complete_axioms le lub" shows "omega_complete le lub"
id2: shows "pCons (rat_of_int (coeff (pCons b c2) (degree q) div lead_coeff q)) (of_int_poly c1) = pCons (coeff (pCons (rat_of_int b) (of_int_poly c2)) (degree q) / coeff (of_int_poly q) (degree q)) (of_int_poly c1)"
bnds_cos_upper: fixes xl :: "float"   and x :: "real"   and xu :: "float"   and y :: "real"   and prec :: "nat" assumes "real_of_float xl \<le> x"   and "x \<le> real_of_float xu"   and "y \<le> cos x" shows "y \<le> real_of_float (snd (bnds_cos prec xl xu))"
assms_1: shows "\<rho> \<in> space (state_measure V' \<Gamma>)"
res_add_eq: fixes x :: "int"   and y :: "int" shows "x \<oplus> y = (x + y) mod m"
closure_subset: fixes A :: "'a set" assumes "A \<subseteq> carrier" shows "closure A \<subseteq> carrier"
minus_divide_left: fixes R :: "('b, 'c) ring_scheme"   and a :: "'b"   and b :: "'b" assumes "field R"   and "a \<in> carrier R"   and "b \<in> carrier R"   and "b \<noteq> \<zero>\<^bsub>R\<^esub>" shows "\<ominus>\<^bsub>R\<^esub> (a \<oslash>\<^bsub>R\<^esub> b) = \<ominus>\<^bsub>R\<^esub> a \<oslash>\<^bsub>R\<^esub> b"
push_bit_double: fixes plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and modulo :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and bit :: "'b \<Rightarrow> nat \<Rightarrow> bool"   and push_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and drop_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and take_bit :: "nat \<Rightarrow> 'b \<Rightarrow> 'b"   and n :: "nat"   and a :: "'b" assumes "class.semiring_bit_shifts plus minus zero times one divide modulo bit push_bit drop_bit take_bit" shows "push_bit n (times a (numeral.numeral one plus (num.Bit0 num.One))) = times (push_bit n a) (numeral.numeral one plus (num.Bit0 num.One))"
sorted_rev_iff_nth_Suc: fixes less_eq :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and less :: "'g \<Rightarrow> 'g \<Rightarrow> bool"   and xs :: "'g list" assumes "class.linorder less_eq less" shows "linorder.sorted less_eq (rev xs) = (\<forall>i. Suc i < length xs \<longrightarrow> less_eq (xs ! Suc i) (xs ! i))"