hcju/isabelleps huggingface dataset:
simps_1: fixes x1 :: "exesignature"   and x2 :: "Core.term list"   and y1 :: "exesignature"   and y2 :: "Core.term list" shows "(ExeTheory x1 x2 = ExeTheory y1 y2) = (x1 = y1 \<and> x2 = y2)"
foldli_not_cond: fixes c :: "'c \<Rightarrow> bool"   and \<sigma> :: "'c"   and xs :: "'d list"   and f :: "'d \<Rightarrow> 'c \<Rightarrow> 'c" assumes "\<not> c \<sigma>" shows "foldli xs c f \<sigma> = \<sigma>"
hyps: fixes P :: "'state" shows "P \<Turnstile> x = P \<Turnstile>\<^sub>S S_transform x"
prems_2: shows "\<forall>aa. ListMem aa (a # as) \<longrightarrow> action_needed_asses aa s2 \<subseteq>\<^sub>f action_needed_asses aa s1"
eoutKnowCorrect_L1k: fixes C :: "specID"   and m :: "Keys" assumes "eoutKnowCorrect C (kKS m)"   and "eout C (kE m)" shows "m \<in> specKeys C \<or> know C (kKS m)"
is_open_subset: fixes X :: "'c set"   and is_open :: "'c set \<Rightarrow> bool"   and U :: "'c set"   and I :: "real set"   and C :: "real \<Rightarrow> 'c set" assumes "open_cover_of_open_subset X is_open U I C" shows "is_open U"
Ln_of_real: fixes z :: "real" assumes "0 < z" shows "ln (complex_of_real z) = complex_of_real (ln z)"
F: shows "Says n run 5 (User m) (Card n) (Crypt (sesK (c * f)) (Passwd m)) \<in> set evsFR5"
psubst_def: fixes \<phi> :: "'fmla"   and txs :: "('trm \<times> 'var) list" shows "psubst \<phi> txs = (let xs = map snd txs; ts = map fst txs; us = getFrN xs ts [\<phi>] (length xs) in rawpsubst (rawpsubst \<phi> (zip (map Var us) xs)) (zip ts us))"
wtE_def: fixes \<xi> :: "Preliminaries.var \<Rightarrow> univ" shows "GE.wtE \<xi> \<equiv> \<forall>x. eintT (GE.tpOfV x) (\<xi> x)"
subexpr_reduct: fixes d1 :: "exp"   and e1 :: "exp"   and CT :: "nat \<Rightarrow> classDef option"   and d2 :: "exp" assumes "d1 \<in> subexprs(e1)"   and "CT \<turnstile> d1 \<rightarrow> d2" shows "\<exists>e2. CT \<turnstile> e1 \<rightarrow> e2"
r_ifz: fixes cond :: "nat"   and val0 :: "nat"   and val1 :: "nat" shows "eval r_ifz [cond, val0, val1] \<down>= (if cond = 0 then val0 else val1)"
narrow_ivl_def: fixes ivl1 :: "ivl"   and ivl2 :: "ivl" shows "ivl1 \<triangle> ivl2 = (case (ivl1, ivl2) of (I l1 h1, I l2 h2) \<Rightarrow> I (if l1 = None then l2 else l1) (if h1 = None then h2 else h1))"
distinct_clearjunk: fixes al :: "('a \<times> 'b) list" shows "distinct (list.map fst (AList.clearjunk al))"
rec_exec_pr_0_simps: fixes n :: "nat"   and f :: "recf"   and g :: "recf"   and xs :: "nat list" shows "rec_exec (Pr n f g) (xs @ [0]) = rec_exec f xs"
Write\<^sub>s\<^sub>b_in_outstanding_refs: fixes a :: "nat"   and sop :: "nat set \<times> ((nat \<Rightarrow> nat option) \<Rightarrow> nat)"   and v :: "nat"   and A :: "nat set"   and L :: "nat set"   and R :: "nat set"   and W :: "nat set"   and xs :: "'b memref list" assumes "Write\<^sub>s\<^sub>b True a sop v A L R W \<in> set xs" shows "a \<in> outstanding_refs is_volatile_Write\<^sub>s\<^sub>b xs"
algebraic_int_times: fixes x :: "'a"   and y :: "'a" assumes "algebraic_int x"   and "algebraic_int y" shows "algebraic_int (x * y)"
inv: shows "2 \<cdot> g \<cdot> x - 2 \<cdot> g \<cdot> h = v \<cdot> v"
A5: shows "strong_reduction_simulation Rel (STCal Source Target) \<and> (\<forall>P Q Q'. (P, Q) \<in> Rel \<and> Q \<longmapsto>STCal Source Target Q' \<longrightarrow> (\<exists>P'. P \<longmapsto>STCal Source Target P' \<and> (P', Q') \<in> Rel))"
is_ground_atm_is_ground_on_var: fixes A :: "('a, 'c) Term.term"   and \<sigma> :: "'c \<Rightarrow> ('a, 'b) Term.term"   and v :: "'c" assumes "is_ground_atm (A \<cdot> \<sigma>)"   and "v \<in> vars_term A" shows "is_ground_atm (\<sigma> v)"
tp_cos: fixes n :: "real" assumes "n \<in> \<nat>" shows "trigpoly (\<lambda>x. cos (n * x))"
mono_invE: fixes f :: "'a \<Rightarrow> 'b"   and x :: "'a"   and y :: "'a" assumes "mono f"   and "f x < f y" obtains "x \<le> y"
conjugation_pair_commute: fixes f :: "'a \<Rightarrow> 'a"   and g :: "'a \<Rightarrow> 'a" assumes "conjugation_pair f g" shows "conjugation_pair g f"
iso_finite: fixes G :: "('a, 'b) monoid_scheme"   and H :: "('c, 'd) monoid_scheme" assumes "G \<cong> H" shows "finite (carrier G) = finite (carrier H)"
env_def: fixes w :: "'a el_step list" shows "env w \<equiv> filter (\<lambda>e. case e of LOC a \<Rightarrow> False | ENV a \<Rightarrow> True) w"
inv_ide: fixes a :: "'a" assumes "ide a" shows "local.inv a = a"
tendsto_infdist: fixes f :: "'b \<Rightarrow> 'c"   and l :: "'c"   and F :: "'b filter"   and A :: "'c set" assumes "(f \<longlongrightarrow> l) F" shows "((\<lambda>x. infdist (f x) A) \<longlongrightarrow> infdist l A) F"
hm_lookup_rule': fixes m :: "'b \<Rightarrow> 'a option"   and l :: "('b \<times> 'a) list list"   and ht :: "('b, 'a) hashtable"   and k :: "'b" shows "<is_hashmap' m l ht> hm_lookup k ht <\<lambda>r. is_hashmap' m l ht * \<up> (r = m k)>"
List_empty: fixes p :: "ref"   and h :: "ref \<Rightarrow> ref" shows "List p h [] = (p = Null)"
s_def: shows "a = s *\<^sub>C (1::'a)"
simps_1: fixes f :: "'a \<Rightarrow> 'f \<Rightarrow> 'g \<Rightarrow> 'e"   and g :: "('a \<times> 'f) list \<Rightarrow> ('a \<times> 'e) list"   and h :: "('a \<times> 'g) list \<Rightarrow> ('a \<times> 'e) list"   and xs :: "('a \<times> 'f) list" shows "map2_val_pair f g h xs [] = g xs"
partial_deriv_one_d_integrable: fixes xc :: "real" assumes "xc \<in> cbox a b" shows "((\<lambda>y. partial_vector_derivative (\<lambda>a. F a \<bullet> i) j (xc, y)) has_integral F (xc, g1 xc) \<bullet> i - F (xc, g2 xc) \<bullet> i) (cbox (g2 xc) (g1 xc))"
B: fixes x :: "hf"   and x' :: "hf"   and x'' :: "hf" assumes "B x x' x''" shows "P x x' x''"
InvariantConflictClauseCharacterizationAfterExhaustivePropagate: fixes state :: "State" assumes "exhaustiveUnitPropagate_dom state"   and "InvariantWatchListsContainOnlyClausesFromF (getWatchList state) (getF state)"   and "InvariantWatchListsUniq (getWatchList state)"   and "InvariantWatchListsCharacterization (getWatchList state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesEl (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantWatchesDiffer (getF state) (getWatch1 state) (getWatch2 state)"   and "InvariantConflictClauseCharacterization (getConflictFlag state) (getConflictClause state) (getF state) (getM state)" shows "let state' = exhaustiveUnitPropagate state in InvariantConflictClauseCharacterization (getConflictFlag state') (getConflictClause state') (getF state') (getM state')"
dec: shows "awalk_cyc_decomp p = (q, r, s)"
index_mult_mat_1: fixes i :: "nat"   and A :: "'c mat"   and j :: "nat"   and B :: "'c mat" assumes "i < dim_row A"   and "j < dim_col B" shows "(A * B) $$ (i, j) = row A i \<bullet> col B j"
max_enat_simps_2: fixes q :: "enat" shows "max q 0 = q"
min_0R: fixes n :: "nat" shows "min n 0 = 0"
positive_proj_psi_l: fixes l :: "nat" shows "positive (proj_psi_l l)"
vsubstEnv_substEnv_compose2: fixes rho :: "'varSort \<Rightarrow> 'var \<Rightarrow> ('a, 'b, 'varSort, 'var, 'c) qTerm set option"   and Y :: "('a, 'b, 'varSort, 'var, 'c) qTerm set"   and ys :: "'varSort"   and zs :: "'varSort"   and y :: "'var"   and z :: "'var"   and z1 :: "'var" assumes "goodEnv rho"   and "good Y"   and "ys \<noteq> zs \<or> y \<notin> {z, z1}" shows "rho &[Y / y]_ys &[z1 // z]_zs = rho &[z1 // z]_zs &[Y #[z1 // z]_zs / y]_ys"
ennreal_of_enat_def: fixes n :: "enat" shows "ennreal_of_enat n = (case n of enat n \<Rightarrow> of_nat n | \<infinity> \<Rightarrow> \<top>)"
stable_pair_on_Cd: fixes ds :: "'b set"   and XD_XH :: "'a set \<times> 'a set"   and d :: "'b" assumes "stable_pair_on ds XD_XH"   and "d \<in> ds" shows "Cd d (fst XD_XH) = match XD_XH \<inter> Field (Pd d)"
of_nat_fun_apply: fixes n :: "nat"   and x :: "'b" shows "of_nat n x = of_nat n"
simps_7: fixes f1 :: "'a"   and f2 :: "'a"   and f3 :: "'a" shows "(case LESS of LESS \<Rightarrow> f1 | EQUAL \<Rightarrow> f2 | GREATER \<Rightarrow> f3) = f1"
cBex_triv_one_point1: fixes A :: "'d cset"   and a :: "'d" shows "cBex A (\<lambda>x. x = a) = cin a A"
ab_semigroup_mult_hom_axioms: shows "ab_semigroup_mult_hom (map_poly of_real)"
sup_finite_5_def: fixes x :: "finite_5"   and y :: "finite_5" shows "x \<squnion> y = (case (x, y) of (a\<^sub>1, a\<^sub>1) \<Rightarrow> a\<^sub>1 | (a\<^sub>1, a\<^sub>2) \<Rightarrow> a\<^sub>2 | (a\<^sub>1, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>1, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>1, a\<^sub>5) \<Rightarrow> a\<^sub>5 | (a\<^sub>2, a\<^sub>3) \<Rightarrow> a\<^sub>3 | (a\<^sub>2, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>2, a\<^sub>5) \<Rightarrow> a\<^sub>5 | (a\<^sub>2, _) \<Rightarrow> a\<^sub>2 | (a\<^sub>3, a\<^sub>4) \<Rightarrow> a\<^sub>5 | (a\<^sub>3, a\<^sub>5) \<Rightarrow> a\<^sub>5 | (a\<^sub>3, _) \<Rightarrow> a\<^sub>3 | (a\<^sub>4, a\<^sub>1) \<Rightarrow> a\<^sub>4 | (a\<^sub>4, a\<^sub>4) \<Rightarrow> a\<^sub>4 | (a\<^sub>4, _) \<Rightarrow> a\<^sub>5 | (a\<^sub>5, xa) \<Rightarrow> a\<^sub>5)"
divide_minus1: fixes divide :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and inverse :: "'b \<Rightarrow> 'b"   and times :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and one :: "'b"   and plus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and zero :: "'b"   and minus :: "'b \<Rightarrow> 'b \<Rightarrow> 'b"   and uminus :: "'b \<Rightarrow> 'b"   and x :: "'b" assumes "class.field divide inverse times one plus zero minus uminus" shows "divide x (uminus one) = uminus x"
congruent_triangleD_4: fixes a1 :: "'b"   and b1 :: "'b"   and c1 :: "'b"   and a2 :: "'c"   and b2 :: "'c"   and c2 :: "'c" assumes "congruent_triangle a1 b1 c1 a2 b2 c2" shows "dist b1 a1 = dist a2 b2"
cases: fixes ground_heads_var :: "'a \<Rightarrow> 'b set"   and gt_sym :: "'b \<Rightarrow> 'b \<Rightarrow> bool"   and arity_sym :: "'b \<Rightarrow> enat"   and arity_var :: "'a \<Rightarrow> enat"   and extf :: "'b \<Rightarrow> (('b, 'a) tm \<Rightarrow> ('b, 'a) tm \<Rightarrow> bool) \<Rightarrow> ('b, 'a) tm list \<Rightarrow> ('b, 'a) tm list \<Rightarrow> bool"   and a1 :: "('b, 'a) tm"   and a2 :: "('b, 'a) tm" assumes "epo ground_heads_var gt_sym arity_sym arity_var extf"   and "epo.gt_chop ground_heads_var gt_sym extf a1 a2" obtains t :: "('b, 'a) tm" and s :: "('b, 'a) tm" where "a1 = t" and "a2 = s" and "is_App t" and "epo.gt ground_heads_var gt_sym extf (chop t) s \<or> chop t = s"
rep_eq: fixes x :: "('d, 'c) sq_matrix" shows "to_fun (Square_Matrix.transpose x) = (\<lambda>i j. to_fun x j i)"
count_roots_interval_sf_rat_def: fixes p :: "int poly" shows "Sturm_Rat.count_roots_interval_sf_rat p = (let pp = real_of_int_poly p; (cr, nr) = count_roots_interval_sf pp in Root_Info (\<lambda>a b. cr (real_of_rat a) (real_of_rat b)) (\<lambda>a. nr (real_of_rat a)))"
fold_equality: fixes z :: "'a"   and B :: "'a set"   and y :: "'a" assumes "fold_graph f z B y"   and "B \<subseteq> A"   and "z \<in> A" shows "Finite_Set.fold f z B = y"
strict_prefix_def: fixes xs :: "'a list"   and zs :: "'a list" shows "(xs < zs) = (xs \<le> zs \<and> \<not> zs \<le> xs)"
formulaEntailsValuation_def: fixes formula :: "Literal list list"   and valuation :: "Literal list" shows "formulaEntailsValuation formula valuation \<equiv> \<forall>literal. literal el valuation \<longrightarrow> formulaEntailsLiteral formula literal"
assign_elim: fixes x :: "'Var"   and e :: "'AExp"   and mds :: "Mode \<Rightarrow> 'Var set"   and mem :: "'Var \<Rightarrow> 'Val"   and c' :: "('Var, 'AExp, 'BExp) Stmt"   and mds' :: "Mode \<Rightarrow> 'Var set"   and mem' :: "'Var \<Rightarrow> 'Val" assumes "\<langle>x \<leftarrow> e, mds, mem\<rangle> \<leadsto> \<langle>c', mds', mem'\<rangle>" shows "c' = Stop \<and> mds = mds' \<and> mem' = mem(x := ev\<^sub>A mem e)"
UnderS_Field: fixes r :: "('a \<times> 'a) set"   and A :: "'a set" shows "UnderS r A \<subseteq> Field r"
less_neg_neg: fixes plus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and zero :: "'a"   and minus :: "'a \<Rightarrow> 'a \<Rightarrow> 'a"   and uminus :: "'a \<Rightarrow> 'a"   and less_eq :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and less :: "'a \<Rightarrow> 'a \<Rightarrow> bool"   and a :: "'a" assumes "class.linordered_ab_group_add plus zero minus uminus less_eq less" shows "less a (uminus a) = less a zero"
partial_magma_axioms: shows "partial_magma (\<cdot>\<^sub>B)"
subgraphE: fixes H :: "('c, 'd) pre_digraph"   and G :: "('c, 'd) pre_digraph" assumes "subgraph H G" obtains "verts H \<subseteq> verts G" and "arcs H \<subseteq> arcs G" and "compatible G H" and "wf_digraph H" and "wf_digraph G"
zhmset_of_less: fixes M :: "hmultiset"   and N :: "hmultiset" shows "(zhmset_of M < zhmset_of N) = (M < N)"
cxt_inv: fixes E :: "('Var, 'AExp, 'BExp) Stmt list"   and c :: "('Var, 'AExp, 'BExp) Stmt"   and c' :: "('Var, 'AExp, 'BExp) Stmt" assumes "cxt_to_stmt E c = c'"   and "\<And>p q. c' \<noteq> p ;; q" shows "E = [] \<and> c' = c"
nsqn_rt_fresh_asI: fixes dip :: "nat"   and rt :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option"   and rt' :: "nat \<Rightarrow> (nat \<times> k \<times> f \<times> nat \<times> nat \<times> nat set) option" assumes "dip \<in> kD rt"   and "dip \<in> kD rt'"   and "nsqn rt dip = nsqn rt' dip"   and "\<pi>\<^sub>5 (the (rt dip)) = \<pi>\<^sub>5 (the (rt' dip))" shows "rt \<approx>\<^bsub>dip\<^esub> rt'"
F4_def: shows "F4 = filter (\<lambda>f. \<not> (\<exists>v\<in>set V3. f \<in> set (facesAt g v))) F2"
assms: shows "(\<gamma>' \<WW>\<^sub>c \<gamma>) t n"
ignore: fixes x :: "'call"   and s :: "'s" assumes "\<not> consider' x" shows "interaction_bounded_by consider (callee s x) 0"
simps_18: fixes f1 :: "nat \<Rightarrow> 'a"   and f2 :: "(val \<times> val) fset \<Rightarrow> 'a"   and f3 :: "val option \<Rightarrow> 'a"   and f4 :: "'a"   and x3 :: "val option" shows "(case Abs x3 of VNat x \<Rightarrow> f1 x | val.Fun x \<Rightarrow> f2 x | Abs x \<Rightarrow> f3 x | Wrong \<Rightarrow> f4) = f3 x3"
linear: fixes var :: "nat"   and L :: "real list"   and C :: "real"   and b :: "real mpoly"   and c :: "real mpoly"   and B :: "real"   and A :: "atom"   and v :: "real" assumes "var < length L"   and "C \<noteq> 0"   and "var \<notin> vars b"   and "var \<notin> vars c"   and "insertion (nth_default 0 (L[var := B / C])) b = B"   and "insertion (nth_default 0 (L[var := B / C])) c = C" shows "aEval A (L[var := B / C]) = aEval (linear_substitution var b c A) (L[var := v])"
int_of_uint8_code: fixes x :: "uint8" shows "int_of_uint8 x = int_of_integer (integer_of_uint8 x)"
a_inv_inj: shows "inj_on (a_inv R) (carrier R)"
a_in_xs: shows "a \<in> set (sorted_list_of_set I)"
add_set_single_nonempty2: fixes f :: "'e \<Rightarrow> 'e \<Rightarrow> 'e"   and z :: "'e"   and i :: "'e \<Rightarrow> 'e"   and a :: "'e"   and n :: "nat" assumes "commute_bpp f (aug_pm_set z i {a})"   and "assoc_bpp (aug_pm_set z i {a}) f"   and "ipp_cond1 {a} i"   and "ipp_cond2 z {a} i f"   and "ipp_cond3 z i"   and "inv_ipp z i f {a}"   and "commute_bpp f (aug_pm_set z i {a})"   and "zeroA z i f {a} z" shows "\<exists>x. x \<in> add_set f {\<^sub>i- a} n"
HSem_eq: fixes ESem :: "'d \<Rightarrow> ('b \<Rightarrow> 'c) \<rightarrow> 'c"   and \<Gamma> :: "('b \<times> 'd) list"   and \<rho> :: "'b \<Rightarrow> 'c" shows "has_ESem.HSem ESem \<Gamma>\<cdot>\<rho> = \<rho> ++\<^bsub>domA \<Gamma>\<^esub> evalHeap \<Gamma> (\<lambda>e. ESem e\<cdot>(has_ESem.HSem ESem \<Gamma>\<cdot>\<rho>))"
Abs_cases: fixes x :: "'c stream" obtains y :: "'c llist" where "x = stream_of_llist y" and "y \<in> {xs. \<not> lfinite xs}"
simps_2: fixes uu :: "'a"   and b :: "'a" shows "the_default uu (Some b) = b"
start_heap_def: fixes G :: "'b prog" shows "start_heap G \<equiv> [addr_of_sys_xcpt NullPointer \<mapsto> blank G NullPointer, addr_of_sys_xcpt ClassCast \<mapsto> blank G ClassCast, addr_of_sys_xcpt OutOfMemory \<mapsto> blank G OutOfMemory, addr_of_sys_xcpt NoClassDefFoundError \<mapsto> blank G NoClassDefFoundError, addr_of_sys_xcpt IncompatibleClassChangeError \<mapsto> blank G IncompatibleClassChangeError, addr_of_sys_xcpt NoSuchFieldError \<mapsto> blank G NoSuchFieldError, addr_of_sys_xcpt NoSuchMethodError \<mapsto> blank G NoSuchMethodError]"
line_integral_exists_smooth: fixes \<gamma> :: "real \<Rightarrow> 'a"   and basis :: "'a set"   and F :: "'a \<Rightarrow> 'a" assumes "\<gamma> C1_differentiable_on {0..1}"   and "\<forall>b\<in>basis. continuous_on (path_image \<gamma>) (\<lambda>x. F x \<bullet> b)"   and "finite basis" shows "line_integral_exists F basis \<gamma>"
assms: shows "InvariantGetReasonIsReason (getReason state) (getF state) (getM state) (set (getQ state))"
Fpow_def: fixes A :: "'a set" shows "Fpow A \<equiv> {X. X \<subseteq> A \<and> finite X}"
Cauchy_iff: fixes X :: "nat \<Rightarrow> 'c" shows "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m\<ge>M. \<forall>n\<ge>M. norm (X m - X n) < e)"
firstPosEqual: fixes a :: "'a"   and l :: "'a list"   and b :: "'a" assumes "a \<in> set l"   and "b \<in> set l" shows "(firstPos a l = firstPos b l) = (a = b)"
JF2rel_F2rel: fixes R :: "'d \<Rightarrow> 'e \<Rightarrow> bool"   and a :: "'d JF2"   and b :: "'e JF2" shows "JF2rel R a b = F2rel R (JF1rel R) (JF2rel R) (dtor2 a) (dtor2 b)"
rotate: shows "ChamberComplexApartmentSystemTriangle X \<A> B' A B D E C z"
graph_Entry_axioms_def: fixes \<alpha>n :: "'a \<Rightarrow> 'b list"   and invar :: "'a \<Rightarrow> bool"   and inEdges' :: "'a \<Rightarrow> 'b \<Rightarrow> ('b \<times> 'c) list"   and Entry :: "'a \<Rightarrow> 'b" shows "graph_Entry_axioms \<alpha>n invar inEdges' Entry \<equiv> (\<forall>g. Entry g \<in> set (\<alpha>n g)) \<and> (\<forall>g. invar g \<longrightarrow> graph_path_base.inEdges inEdges' g (Entry g) = []) \<and> (\<forall>n g. n \<in> set (\<alpha>n g) \<longrightarrow> invar g \<longrightarrow> (\<exists>ns. graph_path_base.path2 \<alpha>n invar inEdges' g (Entry g) ns n))"
extensional_restrict: fixes f :: "'d \<Rightarrow> 'e"   and A :: "'d set" assumes "f \<in> extensional A" shows "restrict f A = f"
cases: fixes a1 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)"   and a2 :: "'Var \<Rightarrow> 'BExp set option"   and a3 :: "'Var set \<times> 'Var set"   and a4 :: "'BExp set"   and a5 :: "(('Var, 'AExp, 'BExp) Stmt \<times> (Mode \<Rightarrow> 'Var set)) \<times> ('Var \<Rightarrow> 'Val)" assumes "a1 \<R>\<^sup>3\<^bsub>a2,a3,a4\<^esub> a5" obtains c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and mds :: "Mode \<Rightarrow> 'Var set" and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val" and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and \<S> :: "'Var set \<times> 'Var set" and P :: "'BExp set" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" and c :: "('Var, 'AExp, 'BExp) Stmt" and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option" and \<S>' :: "'Var set \<times> 'Var set" and P' :: "'BExp set" where "a1 = \<langle>c\<^sub>1__ ;; c__, mds__, mem\<^sub>1__\<rangle>" and "a2 = \<Gamma>''__" and "a3 = \<S>''__" and "a4 = P'__" and "a5 = \<langle>c\<^sub>2__ ;; c__, mds__, mem\<^sub>2__\<rangle>" and "\<langle>c\<^sub>1__, mds__, mem\<^sub>1__\<rangle> \<R>\<^sup>1\<^bsub>\<Gamma>'__,\<S>'__,Pa__\<^esub> \<langle>c\<^sub>2__, mds__, mem\<^sub>2__\<rangle>" and "\<turnstile> \<Gamma>'__,\<S>'__,Pa__ {c__} \<Gamma>''__,\<S>''__,P'__"   | c\<^sub>1 :: "('Var, 'AExp, 'BExp) Stmt" and mds :: "Mode \<Rightarrow> 'Var set" and mem\<^sub>1 :: "'Var \<Rightarrow> 'Val" and \<Gamma> :: "'Var \<Rightarrow> 'BExp set option" and \<S> :: "'Var set \<times> 'Var set" and P :: "'BExp set" and c\<^sub>2 :: "('Var, 'AExp, 'BExp) Stmt" and mem\<^sub>2 :: "'Var \<Rightarrow> 'Val" and c :: "('Var, 'AExp, 'BExp) Stmt" and \<Gamma>' :: "'Var \<Rightarrow> 'BExp set option" and \<S>' :: "'Var set \<times> 'Var set" and P' :: "'BExp set" where "a1 = \<langle>c\<^sub>1 ;; c, mds, mem\<^sub>1\<rangle>" and "a2 = \<Gamma>'" and "a3 = \<S>'" and "a4 = P'" and "a5 = \<langle>c\<^sub>2 ;; c, mds, mem\<^sub>2\<rangle>" and "\<langle>c\<^sub>1, mds, mem\<^sub>1\<rangle> \<R>\<^sup>3\<^bsub>\<Gamma>,\<S>,P\<^esub> \<langle>c\<^sub>2, mds, mem\<^sub>2\<rangle>" and "\<turnstile> \<Gamma>,\<S>,P {c} \<Gamma>',\<S>',P'"
mult_ge_zero: fixes a :: "'c"   and b :: "'c" assumes "(0::'c) \<le> a"   and "(0::'c) \<le> b" shows "(0::'c) \<le> a * b"
Measure_P_def: fixes vars :: "nat set"   and n :: "nat"   and Ps :: "nat \<Rightarrow> complex mat"   and Cs :: "com list" shows "ps_P.Measure_P vars n Ps Cs = Measure n (\<lambda>n. mat_extension ps2_P.dims0 vars (Ps n)) Cs"
nb\<^sub>9_1: shows "\<forall>(v, a)\<in>set (effect_of op). map_of (effect_to_assignments op') (v, a) = Some True"
iT_Plus_cut_le2: fixes I :: "nat set"   and k :: "nat"   and t :: "nat" shows "(I \<oplus> k) \<down>\<le> (t + k) = I \<down>\<le> t \<oplus> k"
linear_span_image: fixes s1 :: "'d \<Rightarrow> 'e \<Rightarrow> 'e"   and s2 :: "'d \<Rightarrow> 'f \<Rightarrow> 'f"   and f :: "'e \<Rightarrow> 'f"   and S :: "'e set" assumes "vector_space_pair s1 s2"   and "linear s1 s2 f" shows "module.span s2 (f ` S) = f ` module.span s1 S"
incrIndexList_help71: fixes m :: "nat"   and nmax :: "nat" shows "\<not> incrIndexList [] m nmax"
ivl_disj_un_two_8: fixes l :: "'a"   and m :: "'a"   and u :: "'a" assumes "l \<le> m"   and "m \<le> u" shows "{l..m} \<union> {m<..u} = {l..u}"
postdominate_implies_path: fixes sourcenode :: "'a \<Rightarrow> 'b"   and targetnode :: "'a \<Rightarrow> 'b"   and kind :: "'a \<Rightarrow> 'c edge_kind"   and valid_edge :: "'a \<Rightarrow> bool"   and Entry :: "'b"   and Exit :: "'b"   and n' :: "'b"   and n :: "'b" assumes "Postdomination sourcenode targetnode kind valid_edge Entry Exit"   and "Postdomination.postdominate sourcenode targetnode valid_edge Exit n' n" obtains as :: "'a list" where "CFG.path sourcenode targetnode valid_edge n as n'"
CRR_hyps_axioms: shows "CRR_hyps p M geom_proc u d init Mkt r risk_free_asset stk"
has_vector_derivative_def: fixes f :: "real \<Rightarrow> 'c"   and f' :: "'c"   and net :: "real filter" shows "(f has_vector_derivative f') net = (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
nat_pow_0: fixes x :: "nat \<Rightarrow> int" shows "x [^]\<^bsub>UP INTEG\<^esub> 0 = \<one>\<^bsub>UP INTEG\<^esub>"
s_inj_2: fixes m :: "nat"   and m' :: "nat" assumes "s2 m = s2 m'" shows "m = m'"
antiderivative_continuous: fixes a :: "real"   and b :: "real"   and f :: "real \<Rightarrow> 'a" assumes "continuous_on {a..b} f" obtains g :: "real \<Rightarrow> 'a" where "\<And>x. x \<in> {a..b} \<Longrightarrow> (g has_vector_derivative f x) (at x within {a..b})"
group_integer_mod_group: fixes n :: "nat" shows "Group.group (Z n)"
create_element_def: shows "create_element \<equiv> l_create_element\<^sub>C\<^sub>o\<^sub>r\<^sub>e\<^sub>_\<^sub>D\<^sub>O\<^sub>M_defs.a_create_element get_disconnected_nodes set_disconnected_nodes set_tag_name"
A_def: shows "A = Matrix.mat 2 2 (\<lambda>(i, j). if i = 0 \<and> j = 0 then a else if i = 0 \<and> j = 1 then b else if i = 1 \<and> j = 0 then 0::'a else c)"
less_sets_imp_list_less: fixes xs :: "'a list"   and ys :: "'a list" assumes "list.set xs \<lless> list.set ys" shows "xs < ys"
distinct_12: fixes x41 :: "'d regex"   and x42 :: "'d regex"   and x2 :: "'d" shows "Times x41 x42 \<noteq> Test x2"
nonneg_lincomb_def: fixes c :: "'a vec \<Rightarrow> 'a"   and Vs :: "'a vec set"   and b :: "'a vec" shows "nonneg_lincomb c Vs b = (lincomb c Vs = b \<and> c ` Vs \<subseteq> {x. (0::'a) \<le> x})"
matrix_has_derivative: fixes x :: "(real, 'a) vec \<Rightarrow>\<^sub>L (real, 'a) vec" shows "((\<lambda>x. matrix (blinfun_apply x)) has_derivative (\<lambda>h. matrix (blinfun_apply h))) (at x)"
lessThan_iff: fixes i :: "'d"   and less :: "'d \<Rightarrow> 'd \<Rightarrow> bool"   and k :: "'d" shows "(i \<in> ord.lessThan less k) = less i k"
sngs_def: shows "sngs = (\<lambda>n. complex_of_real (2 * pi * real n) * \<i>) ` {0<..}"
ord0_leq_def: shows "ord0_leq \<equiv> {(xa, x). ord0_leqp xa x}"
bottom_def: fixes L :: "('c, 'd) gorder_scheme" shows "\<bottom>\<^bsub>L\<^esub> = (SOME x. least L x (carrier L))"
strategyproofness_2: shows "(pmf (sds R39) d < pmf (sds R29) d \<or> pmf (sds R39) d + pmf (sds R39) c < pmf (sds R29) d + pmf (sds R29) c) \<or> pmf (sds R39) d = pmf (sds R29) d \<and> pmf (sds R39) d + pmf (sds R39) c = pmf (sds R29) d + pmf (sds R29) c"